informs14th ® Applied Probability Conference

July 9–11, 2007 Program Monday July 9, 2007 Track 1 Track 2 Track 3 Track 4 Track 5 Track 6 Track 7 Track 8 Track 9 Room CZ 4 CZ 5 CZ 10 CZ 11 CZ 12 CZ 13 CZ 14 CZ 15 CZ 16 9:00am - 9:15am Opening (Room: Blauwe Zaal) 9:15am - 10:15am Plenary - Peter Glynn (Room: Blauwe Zaal) MA Financial Random Fields Rare Event Asymptotic Scheduling Call Centers 1 MDP 1 Retrial Inventory 1 10:45am - 12:15pm Engineering 1 Simulation 1 Analysis 1 Queues Kou Kaj Dupuis Bassamboo / Borst / Koole Feinberg Artalejo Van Houtum Randhawa Wierman Keppo Scheffler Blanchet Lin Gupta Taylor Bispo Machihara Buyukkaramikli DeGuest Ruiz-Medina Glasserman Tezcan Ayesta Jongbloed Van der Laan Nobel Qiu Peng Kaj Juneja Gurvich Wierman Henderson Haijema Shin Timmer Weber Mahmoodi Dupuis Randhawa Winands Koole Feinberg Artalejo Van Houtum 12:45pm - 1.45pm Tutorial Philippe Robert MB Financial Percolation and Simulation 1 Stability of Stoch. Communication Many-server Games 1 Fluid Queues Search 2:00pm - 3:30pm Engineering 2 Related Topics Networks Systems 1 Models 1 Models Schoutens / Van den Berg Henderson Ramanan Choi Armony Economou Adan Klafter Valdivieso Werker Newman Chick Gamarnik Bae Tezcan Economou Dieker Benichou Koch Newman Haas Reiman Kim Jennings Amir Nazarathy Oshanin Scherer Meester Blanchet Williams Park Ward Dube Margolius Eliazar Valdivieso Kurtz Henderson Zachary Roubos Armony Economou Adan Metzler MC Exit Times Interacting Stoch. Prog. Stoch. Netw. & Flow-Level Markov Control Queueing Inventory 2 4:00pm - 5:30pm Particle Systems & Opt. 1 Diffusions Models Processes Models 1 Rolski Van den Berg Gunawan Budhiraja Prabhakar Van Doorn Lelarge Fralix Yao

Pistorius Van der Hofstad Gunawan Ghosh Prabhakar Ferreira Mohamed Kerner Baron De¸bicki Redig Ren Kang Bramson Jaskiewicz´ Yudaeva Ammar Bijvank Palmowski Van den Berg Rubinstein Atar Massoulié Leskelä Cudinaˇ Peköz Yao Rolski Rubinstein Budhiraja Shah Van Doorn Lelarge Fralix Xie 6:00pm - 10:00pm Reception / Dinner (Location: DAF Museum) Contents

1 Welcome Address 5

2 Program Committee and Organizing Committee7

3 Sponsors 9

4 General Information 11

5 Program at a Glance 13

6 Plenaries and Tutorials 15

7 Social Events 19

8 Auditorium Layout 21

9 University Campus Layout 23

10 Sessions 25

11 Abstracts 49

12 Author Index 163

3 CONTENTS

4 1 Welcome Address

Dear Participant,

It is our pleasure to welcome you to the 14th INFORMS Applied Probability Conference, to be held in the Auditorium of Eindhoven University of Technology. Traditionally, emphasis in this conference is on applications of probability in operations research; typical OR subjects are optimization of logistic networks under uncertainty, control of inventory systems, and per- formance analysis of communication networks. The analysis of the underlying probability models relies predominantly on the theory of Markov chains, stochastic processes, queueing theory, and advanced simu- lation techniques. Due to the strong methodological connections, however, also various other applications are included in the conference, for instance those inspired by finance, physics and biology. We are very pleased that the following researchers, leaders in our field, have agreed to give keynote lectures: Peter Glynn, Frank Kelly and Alain-Sol Sznitman. Another exciting prospect is formed by three tutorials on timely subjects, to be presented by Maury Bramson, Paul Glasserman and Philippe Robert. Furthermore, about 350 twenty-minute talks (roughly as many as the number of registered participants) will be presented in 90 sessions. Each time 9 sessions will run in parallel. We are grateful to the many persons who have assisted us in organizing the conference. In particular, we’d like to mention the Congress Office of Eindhoven University of Technology, the extremely able and helpful staff of EURANDOM, Josine Bruin, Hannie Horvath, Wim Senden, Erik Winands, and last but not least Marko Boon, whose wizardry made the preparation of this Program book almost look easy. We have chosen a schedule that leaves you with ample time to renew acquaintances, to make new friends, and to discuss problems and new ideas. We wish you a pleasant stay in Eindhoven!

July 2007 The organizing committee

5 1 – Welcome Address

6 2 Program Committee and Organizing Committee

Co-Chairs

Onno Boxma (EURANDOM and TU Eindhoven, Eindhoven, The Netherlands, [email protected]) Michel Mandjes (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands, [email protected])

Program Committee

Mor Armony (New York University, New York, USA) Jesus Artalejo (Complutense University of Madrid, Madrid, Spain) Rami Atar (Technion, Haifa, Israel) Francois Baccelli (INRIA & ENS, Paris, France) Sem Borst (Lucent, Murray Hill, USA & TU Eindhoven, Eindhoven, The Netherlands) Richard Boucherie (Twente University, Enschede, The Netherlands) B.D. Choi (Korea University, Seoul, Korea) Neil O’Connell (University of Cork, Cork, Ireland) Dee Denteneer (Philips Research, Eindhoven, The Netherlands) Doug Down (McMaster University, Hamilton, Canada) Paul Dupuis (Brown University, Providence, USA) Serguei Foss (Heriot-Watt University, Edinburgh, UK) Geoffrey Grimmett (University of Cambridge, Cambridge, UK) Peter Glynn (Stanford University, Stanford, USA) Shane Henderson (Cornell University, Cornell, USA) Remco van der Hofstad (TU Eindhoven & EURANDOM, Eindhoven, The Netherlands) Ingemar Kaj (Uppsala University, Uppsala, Sweden) Offer Kella (Hebrew University, Jerusalem, Israel) Ger Koole (Vrije Universiteit Amsterdam, Amsterdam, The Netherlands) Dirk Kroese (University of Queensland, Brisbane, Australia) Thomas Mikosch (University of Copenhagen, Copenhagen, Denmark) Ilkka Norros (VTT, Helsinki, Finland) Balaji Prabhakar (Stanford University, Stanford, USA) Kavita Ramanan (Carnegie Mellon University, Pittsburgh, USA) Ulrich Rieder (University Ulm, Ulm, Germany) Wim Schoutens (KU Leuven, Leuven, Belgium) Hanspeter Schmidli (University of Cologne, Cologne, Germany) Mark Squillante (IBM Thomas J. Watson Research Center, Yorktown Heights, USA) Tetsuya Takine (Osaka University, Osaka, Japan)

7 2 – Program Committee and Organizing Committee

Peter Taylor (University of Melbourne, Melbourne, Australia) Henk Tijms (Vrije Universiteit Amsterdam, Amsterdam, The Netherlands) Assaf Zeevi (Columbia University, New York, USA)

Organizing Committee

Ivo Adan (EURANDOM and TU Eindhoven, Eindhoven, The Netherlands) Onno Boxma (EURANDOM and TU Eindhoven, Eindhoven, The Netherlands) Lucienne Coolen (EURANDOM, Eindhoven, The Netherlands) Patty Koorn (EURANDOM, Eindhoven, The Netherlands) Michel Mandjes (Korteweg-de Vries Institute for Mathematics and University of Amsterdam, Amsterdam, The Netherlands) Jacques Resing (TU Eindhoven, Eindhoven, The Netherlands)

8 3 Sponsors

The organizers gratefully acknowledge the support of the following organizations:

Stichting ABE

Co-sponsored by the INFORMS Applied Probability Society and The Institute of Mathematical Statistics. informs ®

9 3 – Sponsors

10 4 General Information

Location and Conference Buildings

The conference will take place at the Auditorium of Eindhoven University of Technology (see maps in Section8 and9).

(Pre-)Registration and Conference Office

On Sunday July 8, the Conference Office is located in EURANDOM, which is housed in the Laplace Building (LG on the map in Section9). It will be open from 5:00pm till 7:00pm for pre-registration. Refreshments will be served during this meet and greet session. On Monday, Tuesday and Wednesday the Conference Office is located on the first floor of the Auditorium (see map in Section8). On Monday it is open for registration from 7:30am until 6:00pm. On Tuesday and Wednesday it is open from 8:00am until 6:00pm.

Badge

You are kindly requested to wear your key cord and badge during the conference as it grants you access to the buildings and use of facilities such as catering, DAF-Museum, film and sports.

Computer Facilities

Wireless network access is available at several places in the Auditorium opposite the lecture rooms. You can plug in your notebook and get access with the personal login and password that you received, when you registered beforehand for this facility. If you had not registered beforehand, fill in the form “Wire- less Internet Access” in your conference documentation and deliver it at the reception on Monday after 11:00am. If you don’t have a notebook with wireless connection possibility, you may use the EURAN- DOM notebooks, which are available in lecture room CZ 9 (see map in Section8).

Refreshments and Meals

Coffee, tea and refreshments will be served during the breaks and before the first session of the day near the book stands on Level 1 (see map in Section8). Lunch is being offered in the Senaatszaal on Level 1, just opposite the book stands of Cambridge University Press and Springer. They are included in the registration fee.

11 4 – General Information

Parking

There are several car parks around the Auditorium. One needs a parking ticket to enter and another one to leave the University premises. Exit tickets can be obtained for free at the registration desk, when you hand in your entrance ticket.

Sports Facilities

Eindhoven University of Technology has a very well equipped sports center . INFORMS-participants who are interested in using some of these facilities are invited to visit the sports center (it is building SP on the campus map in Section9). Your congress badge, to be shown at the reception desk, grants you access to the building and facilities. The swimming pool is open daily for free swimming from 7:45am - 8:45am and from 12:00pm - 1:30pm. In the afternoon opening hours are: July 9th Aqua jogging 5:00pm – 6:30pm, participation possible; if there are not enough participants, free swimming is possible. July 10th Free swimming 4:30pm – 6:30pm. July 11th Aqua jogging 5:00pm – 6:30pm, participation possible; if there are not enough participants, free swimming is possible. Gyms for cardio fitness, fitness and other facilities are open from 8:30am until 7:00pm. Use of the facilities at the sports center is at your own risk.

Applied Probability Society Business Meeting

Time and date: Tuesday July 10, 6:15pm – 7:00pm. Location: CZ 4, Auditorium.

12 5 Program at a Glance

Monday July 9, 2007 Session Rooms

9:00am – 09:15am Opening Opening and Plenary 9:15am – 10:15am Plenary - Peter Glynn Blauwe Zaal 10:45am – 12:15pm Parallel Sessions MA Track 1, Tutorials and 12:45pm – 1:45pm Tutorial - Philippe Robert APS Business Meeting 2:00pm – 3:30pm Parallel Sessions MB CZ 4 4:00pm – 5:30pm Parallel Sessions MC 6:00pm – 10:00pm Reception / Dinner DAF Museum Track 2 CZ 5 Tuesday July 10, 2007 Track 3 CZ 10 8:30am – 10:00am Parallel Sessions TA Track 4 10:30am – 12:00pm Parallel Sessions TB CZ 11 12:20pm – 1:20pm Tutorial - Paul Glasserman 1:30pm – 2:30pm Plenary - Alain-Sol Sznitman Track 5 2:45pm – 4:15pm Parallel Sessions TC CZ 12 4:45pm – 6:15pm Parallel Sessions TD Track 6 6:15pm – 7:00pm APS Business Meeting CZ 13 6:30pm – 8:00pm Social Event Track 7 Wednesday July 11, 2007 CZ 14 Track 8 8:30am – 10:00am Parallel Sessions WA CZ 15 10:30am – 12:00pm Parallel Sessions WB Track 9 12:20pm – 1:20pm Tutorial - Maury Bramson CZ 16 1:30pm – 3:00pm Parallel Sessions WC 3:15pm – 4:15pm Plenary - Frank Kelly

13 5 – Program at a Glance

14 6 Plenaries and Tutorials

Monday 9:15am – 10:15am Plenary in Blauwe Zaal

Recent Developments in Rare-Event Simulation

Peter W. Glynn, Stanford University, Stanford, USA Computing rare-event probabilities is a problem that is of interest in many applied settings: loss in com- munications networks, bankruptcy in insurance risk, long customer fulfillment times in a supply chain, valuing an out-of-the money option, assessing system unavailability, etc. In this talk, we will describe recent progress in developing efficient algorithms for computing such rare-event probabilities. Importance sampling plays a key role, specifically the use of changes-of-measure that are "state-dependent" or "dy- namic". Appropriately designed Lyapunov functions are then needed to bound the variances that arise in the analysis of such algorithms. In fact, the Lyapunov bound itself can play a role in the design of the algorithm. This talk describes joint work with Jose Blanchet that is closely related to the Isaacs equation ideas introduced by Dupuis and Wang.

Monday 12:45pm – 1:45pm Tutorial in CZ 4

A Survey on Tree Algorithms and their Analysis

Philippe Robert, INRIA-Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay, France, [email protected]

A tree algorithm is a procedure that recursively divides (splits) an initial set of n items into subsets until each of the subsets obtained has a cardinality strictly less than some fixed number D. These algorithms have a wide range of applications:

• access protocols in communication networks • divide and conquer algorithms for data structures • algorithms in distributed systems (leader election) • statistical tests.

On the mathematical side, the analysis of tree algorithms is connected to classical problems of several research domains such as:

• fragmentation processes • random recursive decomposition

15 6 – Plenaries and Tutorials

• dynamical systems (Zeta functions)

• physics (complex dimensions of strings)

In this talk, we present a general survey of this area and present the mathematical methods used to analyze the asymptotic behavior of additive functionals associated to these algorithms. Several approaches for this analysis will be discussed: Complex Analysis, Functional Operators and Probabilistic Methods (with an emphasis on the latter one). The key result at the heart of the probabilistic analysis is the renewal theorem and its extensions when inde- pendence assumptions are relaxed. The purpose of this talk is also to stress the power and the complexity of this set of results in applied probability theory.

Tuesday 12:20pm – 1:20pm Tutorial in CZ 4

Monte Carlo Problems in Finance

Paul Glasserman, Columbia Business School Uris Hall 3022 Broadway New York, NY 10027, USA This tutorial will discuss some applications and research problems arising in the use of Monte Carlo sim- ulation in finance. I will focus on two problems, in particular, that are of both practical and theoretical interest: the estimation of price sensitivities ("Greeks") for hedging, and the pricing of American options. For the problem of estimating sensitivities, I will first review the pathwise method and the likelihood ratio method, highlighting their strengths and weaknesses for various types of financial problems. A more recent line of work (starting with Fournie et al. 1999) derives estimators using Malliavin calculus. I will discuss connections (based on joint work with Nan Chen) between the Malliavin estimators and the more traditional methods. In several cases, one may arrive at Malliavin estimators by combining the pathwise and likelihood ratio methods in a discrete-time approximation and then passing to the continuous-time limit. An American option grants the holder the right to choose the time of exercise, so pricing an American option entails solving an optimal stopping problem. I will review methods for approximating the prices of American options in complex models by simulation. I will give particular attention to the dual formulations of Rogers (2001), Haugh and Kogan (2004) and Jamshidian (2003), which replace maximization over stopping times with minimization over martingales. I will compare additive and multiplicative forms of duality, based on joint work with Nan Chen: we establish an equivalence in the quality of the bounds achieved by the two methods, but show that the variance of the multiplicative method is typically much larger.

Tuesday 1:30pm – 2:30pm Plenary in Blauwe Zaal

Random Walks and Interfaces

Alain-Sol Sznitman, ETH, Departement Mathematik, Rämistrasse 101 8092 Zürich, Switzerland The general theme of this talk brings together random walks and percolation. We present results related to the way paths of random walks can create interfaces. Such questions arise for instance when investigating the disconnection time by simple random walk of a discrete cylinder based on a large discrete torus (the problem of the "termite in the wooden beam"), or when studying the percolative properties of the vacant set left by simple random walk on a large discrete torus by times comparable to the number of sites in the torus. We also discuss a model describing the microscopic structure "in the bulk" in both examples. Part of the results presented here have been obtained in collaboration with A. Dembo or I. Benjamini.

16 Wednesday 12:20pm – 1:20pm Tutorial in CZ 4

Application of Fluid Limits to Queueing Networks and Associated Lyapunov Func- tions

Maury Bramson, University of Minnesota, Minneapolis, USA Over the past decade, fluid limits and fluid models have become important tools for demonstrating the sta- bility of multiclass queueing networks. Fluid limits are, in essence, the deterministic limits of the queueing network process obtained under law of large number scaling, and the corresponding fluid model is the set of equations satisfied by such limits. For a large family of disciplines, stability of the fluid model will imply stability of the corresponding queueing network. This connection was first employed in works by Rybko, Stolyar, and, in particular, Dai in the early 1990’s. It is heavily used today. The deterministic framework provided by fluid limits is easier to analyze than that of the corresponding queueing networks. In some cases, Lyapunov functions have been derived; under certain settings, these are given by entropy functions. There is, however, no systematic procedure for analyzing fluid limits. More- over, although the framework considered in Dai applies to a large family of queueing networks, various recent networks of interest lie outside this family. Here, we review the basic theory of fluid limits and give specific examples of their application. We also present certain networks where the current theory of fluid limits does not apply, and discuss alternative approaches to these problems.

Wednesday 3:15pm – 4:15pm Plenary in Blauwe Zaal

Models of routing and congestion control

Frank Kelly, University of Cambridge, Cambridge, UK The Internet has attracted the attention of many theoreticians, eager to understand the remarkable success of this diverse and complex artefact. One strand of this effort has been a framework that allows a congestion control algorithm such as Jacobson’s TCP to be interpreted as a distributed mechanism solving a global optimization problem. The framework is based on fluid models of packet flows, and the form of the optimization problem makes explicit the equilibrium resource allocation policy of the algorithm, which can often be restated in terms of a fairness criterion, such as proportional fairness or max-min fairness. The dynamics of the fluid models allow the machinery of control theory to be used to study stability, and to develop joint routing and rate control algorithms. A further strand of theoretical work has developed a connection level model, that represents the randomly varying number of flows present in a network. Stability on this time-scale is interpreted in terms of recurrence of a Markov chain. Under proportional fairness and a local traffic condition, recent work with Kang, Lee and Williams has developed a heavy traffic diffusion approximation under which the workload process behaves like Brownian motion in the interior of a polyhedral cone and is confined to the cone by reflection at the boundary. The diffusion has a product form invariant distribution with a strikingly simple interpretation in terms of dual random variables, one for each of the resources of the network. When routing is incorporated, the product form distribution has a dual random variable for each of a set of generalised cut constraints.

17 6 – Plenaries and Tutorials

18 7 Social Events

Film, Monday July 9, 1:00pm – 1:50pm

THE PLEASURE OF FINDING THINGS OUT is fifty minutes of PURE Feynman! This is the original Horizon Nova interview filmed in 1981. It will delight and inspire anyone who would like to share something of the joys of scientific discovery. Feynman is a master storyteller, and his tales – about childhood, Los Alamos, or how he won a Nobel Prize – are a vivid and entertaining insight into the mind of a great scientist at work and play.

Location: the “Zwarte Doos” (ZD on the map in Section9), filmzaal 1.02.

Reception at DAF-Museum, Monday July 9, 6:00pm – 10:00pm

After the last session, which ends at 5:30pm, we get together in front of the Auditorium (see campus map). Guides (with Grey EURANDOM T-shirt) will escort attendees and guests to the DAF-Museum (more information in the flyer in the conference bag).

Hot and cold dishes are going to be served in an attractive atmosphere.

Your congress badge, to be shown at the entrance, will grant you access. Route description for the 20-minute walk to the DAF-Museum:

• Walk in the direction of the railway station, follow for a short while the concrete Limbopath (com- bined cycle path and pedestrian path), take the first path on the left, just after the restaurant/movie theatre De Zwarte Doos (do not cross the Dommel river), go through the two tunnels until the Fuut- laan, cross the Fuutlaan, go more or less straight ahead (Parklaan), take the first street right (is also called Parklaan) and cross the bridge.

• Just after the bridge on the left starts the path called Pad Marienhage. This path is part of the Industrial Walk (Fabrieksroute, a walk along former industrial sites of Eindhoven). You will find a couple of plates (nrs. 16, 15, 14, 13 and 12) with a description (in Dutch) of these former sites. 16. Linen factory Van den Briel & Verster (now a youth centre called The Effenaar, including a restaurant) 15. Linen weaving mill Ignatius de Haas (became part of the Dutch Textile Association in 1965) 14. Soap-making factory/Soap works De Lelie (until 1968) 13. Mock-velvet factory Schellens & Marto (moved to another factory site in Eindhoven) 12. First gas plant of Eindhoven (until the introduction of natural gas in 1965)

19 7 – Social Events

• Follow the path until the Kanaalstraat, turn left, cross the bridge and follow the Kanaaldijk Zuid until the next bridge, turn left, cross the bridge and follow the Tongelresestraat; on the left (almost immediately) you will find the DAF-Museum.

Music and Get Together, Tuesday July 10, 6:30pm – 8:00pm

All participants are warmly invited to come to the open field in front of the Auditorium to enjoy some drinks and snacks. In case of bad weather (after all, this is the Netherlands!), we will meet in the Auditorium. Live music will be provided by the NEW ORLEANS DIXIE SKATERS (www.nods.nl).

20 8 Auditorium Layout

Opening hours: Monday 7:00am – 6:30pm, Tuesday 7:30am – 8pm (or 9pm in case the social event takes place inside the Auditorium) and Wednesday 7:30am – 6:30pm. Lecture rooms CZ 4 and CZ 5 are at level 0. The registration, plenary lectures, lunches and book stands are at level 1. Lecture rooms CZ 9 – CZ 16 and computer facilities are at level 2.

Level 0

Track2

CZ5 CZ6 CZ7 CZ8

CZ4 CZ3 CZ2 CZ1

Track1and Tutorials

21 8 – Auditorium Layout

Level 1

BookstandsRegistrationdesk

Voorhof

Senaatszaal

Blauwe zaal

Lunch OpeningandPlenary

Level 2

CZ13CZ14CZ15CZ16 Track6 Track7 Track8 Track9

Blauwe zaal

CZ12CZ11CZ10 CZ9 Track5 Track4 Track3

ComputerFacilities

22 9 University Campus Layout

SportsCentre

LaplaceGebouw(EURANDOM,

RAILWAY &BUS-STATION Pre-registration) Auditorium(ConferenceSite) ZwarteDoos(Movie Theatre)

AUD: Auditorium (Conference site) ZD: Zwarte Doos (Movie theatre) SP: Sports center LG: Laplace gebouw (EURANDOM, Pre-registration on Sunday)

23 9 – University Campus Layout

24 10 Sessions

Monday 10:45am - 12:15pm

Session MA01 - Financial Engineering 1, Chair: Kou in CZ 4, see page 49

1. Optimal Consumption and Portfolio Management with Non-Observable Real Prices Bensoussan, A.; Keppo, J.; Sethi, S.P.

2. Robustness and sensitivity analysis of risk measurement procedures Cont, R.; DeGuest, R.; Scandolo, G.

3. What Is a Good Risk Measure: Bridging the Gaps between Robustness, Subadditivity, and Insurance Risk Measures Heyde, C.C.; Kou, S.G.; Peng, X.H.

4. Dynamic importance sampling for credit baskets Weber, S.; Liese, M.

Session MA02 - Random Fields, Chair: Kaj in CZ 5, see page 50

1. Operator Scaling Stable Random Fields: Theory and Simulation Scheffler, H.P.; Biermé, H.; Meerschaert, M.M.

2. Random fields generated from multifractional pseudodifferential operators: Pseudostable random fields Ruiz-Medina, M.D.

3. Self-similar random fields as scaling limits of Poisson germ-grain models Kaj, Ingemar; Estrade, Anne; Biermé, H.

4. Characterization of Banach Valued Stable Random Measures Mahmoodi, S.; Soltani, A.R.

Session MA03 - Rare Event Simulation 1, Chair: Dupuis in CZ 10, see page 52

1. Fast Simulation of Brownian Motion Avoiding Poissonian Obstacles Blanchet, Jose; Dupuis, Paul

2. Fast pricing of basket default swaps Chen, Zhiyong; Glasserman, Paul

3. Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Random Variables Glasserman, Paul; Juneja, Sandeep

25 10 – Sessions

4. Design and analysis of splitting algorithms for rare event estimation Dupuis, Paul

Session MA04 - Asymptotic Analysis 1, Chair: Bassamboo / Randhawa in CZ 11, see page 53

1. Heavy Traffic Analysis of Maximum Pressure Policies for Stochastic Processing Networks with Multiple Bottlenecks Ata, Baris; Lin, Wuqin 2. Slow-server problem in many-server heavy traffic with non-exponential service times Tezcan, Tolga 3. Validity of heavy-traffic steady-state approximations in open queueing networks: sufficient condi- tions involving state-space collapse Gurvich, I.; Zeevi, Assaf 4. Rental duration distributions and stock-levels: Why deadlines may be bad for rental firms Bassamboo, A.; Kumar, Sunil; Randhawa, R.S.

Session MA05 - Scheduling, Chair: Borst / Wierman in CZ 12, see page 54

1. New Approximations and Bounds for an M/G/s FIFO queueing system Gupta, Varun; Dai, Jim; Harchol-Balter, Mor; Zwart, Bert 2. Optimal scheduling discipline in an M/G/1 queue with Pareto-type service time distributions Aalto, S.; Ayesta, U. 3. Revisiting the performance of large jobs in the M/GI/1 Wierman, Adam 4. Scheduling in polling systems Winands, E.M.M.; Wierman, Adam; Boxma, O.J.

Session MA06 - Call Centers 1, Chair: Koole in CZ 13, see page 56

1. Density Forecasting of Intraday Call Center Arrivals using Models Based on Exponential Smoothing Taylor, J.W.; Snyder, R.D. 2. Short term dynamic prediction in Call Centers Jongbloed, G.; Eveleens, J.L. 3. Forecast errors in service systems Steckley, Samuel G.; Henderson, Shane G.; Mehrotra, Vijay 4. Shift Scheduling in Multi-skill Call Centers Koole, G.

Session MA07 - MDP 1, Chair: Feinberg in CZ 14, see page 57

1. The entangled scheduling and routing problem Bispo, C.F. 2. The power of deterministic open-loop control policies Van der Laan, D.A.

26 3. A MDP approach to the control of traffic lights Haijema, R.; Van der Wal, Jan

4. Splitting of randomized stationary policies in absorbing Markov Decision Processes Denardo, E.V.; Feinberg, E.A.; Rothblum, U.G.

Session MA08 - Retrial Queues, Chair: Artalejo in CZ 15, see page 59

1. Cellular mobile radio telephone networks with priority active customers Machihara, F.

2. A discrete-time queueing model with abandonments Nobel, R.

3. M/M/s Retrial Queue with Balking and Reneging Shin, Yang Woo; Choo, Taek Sik

4. Algorithmic analysis of the Geo/Geo/c retrial queue Artalejo, J.R.

Session MA09 - Inventory 1, Chair: Van Houtum in CZ 16, see page 60

1. Coordinated replenishment in a two echelon inventory system with transportation capacity Buyukkaramikli, N.C.; Gurler, U.; Alp, O.

2. A multi-item inventory system with multiple resource constraints Qiu, Minmin; Ding, Hongwei; Dong, Jin; Ren, Changrui; Wang, Wei

3. Coordination in divergent multi-echelon inventory systems Timmer, J.

4. A product-form solution for two-echelon spare parts networks with emergency shipments Van Houtum, G.J.; Boucherie, R.J.; Heideveld, S.A.

Monday 2:00pm - 3:30pm

Session MB01 - Financial Engineering 2, Chair: Schoutens / Valdivieso in CZ 4, see page 62

1. Optimal Portfolio Choice with Annuitization Koijen, Ralph S.J.; Nijman, Theo E.; Werker, Bas J.M.

2. Measuring comonotonicity in random vectors Koch, I.

3. A tractable multi-firm default model with jumps Scherer, Matthias

4. Parameter estimation in fractionally integrated process of Ornstein-Uhlenbeck type Valdivieso, Luis

27 10 – Sessions

Session MB02 - Percolation and Related Topics, Chair: Van den Berg in CZ 5, see page 63

1. Exceptional times for the dynamical discrete web, part I Fontes, L.R.G.; Newman, C.M.; Schertzer, E.; Ravishankar, K. 2. Exceptional times for the dynamical discrete web, part II Fontes, L.R.G.; Newman, C.M.; Schertzer, E.; Ravishankar, K. 3. Sharp phase transitions in divide-and-colour models Meester, R.J.; Balint, A.; Camia, Federico 4. Multiscale analysis of stochastic reaction networks Kurtz, Thomas G.

Session MB03 - Simulation 1, Chair: Henderson in CZ 10, see page 64

1. Bandits and simulation selection options Chick, S.E.; Gans, N. 2. On transience and recurrence in discrete-event simulations Haas, P.J.; Glynn, P.W. 3. Analysis Of State-Dependent Importance Sampling Algorithms Blanchet, Jose 4. Adaptive control variates for steady-state simulation Kim, Sujin; Henderson, Shane G.

Session MB04 - Stability of Stochastic Networks, Chair: Ramanan in CZ 11, see page 65

1. Steady-state analysis of a multi-server queueing system in QED regime Gamarnik, D.; Momcilovic, P. 2. The heavy traffic limit of an unbalanced generalized processor sharing model Ramanan, K.; Reiman, Martin I. 3. Fluid model for a data network with alpha fair bandwidth sharing and general document size distri- butions: two examples of stability Gromoll, H.C.; Lee, N.H.; Williams, R.J. 4. Stability of processor sharing networks with simultaneous resource requirements Hansen, J.; Reynolds, C.; Zachary, S.

Session MB05 - Communication Systems 1, Chair: Choi in CZ 12, see page 66

1. Performance Analysis of IEEE 802.11e EDCA in Non-saturation Condition by M/G/1 Queue Bae, Y.H.; Kim, Kyung Jae; Choi, Bong Dae 2. Performance Analysis of IEEE 802.15.4 Unslotted CSMA/CA Kim, Tae Ok; Park, Jin Soo; Kim, Kyung Jae; Choi, Bong Dae 3. Performance analysis of IEEE 802.15.4 Non-beacon Mode by Piggyback Method Park, Jin Soo; Kim, Tae Ok; Kim, Kyung Jae; Choi, Bong Dae 4. Performance Analysis of Session-Level Load Balancing Algorithms Roubos, D.; Bhulai, S.; Van der Mei, R.D.

28 Session MB06 - Many-server Models 1, Chair: Armony in CZ 13, see page 68

1. Optimal Control of Parallel Server Systems with Many Servers Dai, Jim; Tezcan, Tolga 2. Fill rate constraint compliance and the many-machine repairman problem Jennings, O.; Degrimenci, I.T. 3. Fair Dynamic Routing Policies in Large-Scale Service Systems with Heterogeneous Servers Ward, A.R.; Armony, Mor 4. Large cross selling in call centers Gurvich, I.; Armony, Mor; Maglaras, Constantinos

Session MB07 - Games 1, Chair: Economou in CZ 14, see page 69

1. Equilibrium customer strategies under various levels of information for the single server queue with unreliable server. Economou, A.; Kanta, S. 2. On the use of quasi birth-and-death processes in equilibrium selection in game theory and queueing networks Amir, Madjid 3. An analysis of oligopolistic competitive markets for queued services with QoS guarantees Dube, Parijat; Jain, Rahul; Touati, Corinne 4. Optimal balking strategies in the single server Markovian queue with compartmented waiting space. Economou, A.; Kanta, S.

Session MB08 - Fluid Queues, Chair: Adan in CZ 15, see page 70

1. On the stationary measure of a reflected Brownian motion in a wedge: some explicit results Dieker, A.B.; Moriarty, J. 2. Transient Fluid Solutions and Queueing Networks with Infinite Virtual Queues Nazarathy, Y.; Weiss, G. 3. R, G and U and the time-inhomogeneous periodic Quasi-Birth Death Process Margolius, B.H. 4. Insurance Risk with Variable Number of Policies Adan, I.J.B.F.; Kulkarni, Vidyadhar

Session MB09 - Search Models, Chair: Klafter in CZ 16, see page 71

1. Intermittent search processes Benichou, Olivier 2. Search research Oshanin, Gleb 3. Parallel search of long circular strands: Modeling, analysis, and optimization Eliazar, Iddo 4. Stochasticity and search in gene regulation processes Metzler, Ralf

29 10 – Sessions

Monday 4:00pm - 5:30pm

Session MC01 - Exit Times, Chair: Rolski in CZ 4, see page 73

1. On an exit problem of a two-dimensional risk process from the positive quadrant Avram, Florin; Palmowski, Z.; Pistorius, Martijn 2. Gaussian storage processes: long-time behavior De¸bicki, K. 3. De Finetti’s dividend problem for a general Lévy insurance risk process Palmowski, Z.; Avram, Florin; Kyprianou, A.; Pistorius, Martijn 4. The exact asymptotics of the time to collision tail distribution Puchała, Z.; Rolski, T.

Session MC02 - Interacting Particle Systems, Chair: Van den Berg in CZ 5, see page 74

1. The super-process limit of the critical spread-out contact process above 4 spatial dimensions Van der Hofstad, R. 2. Duality and exact correlations in a model of heat conduction Giardina, C.; Kurchan, J.; Redig, F. 3. Approximate zero-one laws, spin-flip dynamics and Ising percolation Van den Berg, J.

Session MC03 - Stochastic Programming and Optimization 1, Chair: Gunawan in CZ 10, see page 75

1. Performance Evaluation of Interconnection Network System Based on Minimum Cut Sets Method Gunawan, Indra 2. Scheduling assembly flow-shops with resource leveling: a GA based heuristic approach Wang, Qinhua; Ren, Changrui; Dong, Jin; Ding, Hongwei; Wang, Wei; Qiu, Minmin 3. A Fast Entropy Method for Solving Integer Programs: Part I Rubinstein, R.Y. 4. A Fast Entropy Method for Solving NP-Hard Counting Problems: Part II Rubinstein, R.Y.

Session MC04 - Stochastic Networks and Diffusions, Chair: Budhiraja in CZ 11, see page 76

1. Optimal buffer size for a stochastic processing network with a drift Ghosh, A.P.; Weerasinghe, A. 2. Characterizations of the invariant measure for a class of reflected diffusions via the extended Sko- rokhod map Kang, Weining; Ramanan, K. 3. Diffusion limit for dynamic routing with random service rates Atar, R.

30 4. Convergence of Invariant Measures for Generalized Jackson Networks Budhiraja, A.; Lee, Chihoon

Session MC05 - Flow-Level Models, Chair: Prabhakar in CZ 12, see page 78

1. Using the Cavity Method to Analyze Randomized Load Balancing Algorithms with General Flow- size Distributions Lu, Y.; Prabhakar, B. 2. Stability of Randomized Load Balancing Algorithms with General Flow-Size Distributions Bramson, M. 3. Proportional fairness: stability and insensitivity properties Massoulié, Laurent 4. Stochastic Stability Under Network Utility Maximization: General File Size Distribution Chiang, M.; Tang, A.; Shah, D.

Session MC06 - Markov Processes, Chair: Van Doorn in CZ 13, see page 79

1. Level-crossing ordering of semi-Markov jump processes and applications to Whittle Networks Ferreira, F.; Pacheco, A. 2. Average optimality for risk-sensitive control with general state space Jaskiewicz,´ A. 3. Stochastic binary relations of random variables and processes Leskelä, L. 4. Survival in a quasi-death process Van Doorn, E.A.

Session MC07 - Control, Chair: Lelarge in CZ 14, see page 80

1. A probabilistic analysis of a leader election algorithm Mohamed, H. 2. Investigation of discrete stochastic systems with dynamic distributed routing and delayed information Yudaeva, N.V. 3. On asymptotic optimality of controls for time-varying queues Cudina, M.; Ramanan, K.; Shreve, S. 4. Heavy traffic limit for online bandwidth packing Lelarge, Marc

Session MC08 - Queueing Models 1, Chair: Fralix in CZ 15, see page 81

1. The conditional distribution of the residual service time in the Mn/G/1 queue Kerner, Yoav 2. Transient Solution to the Time Dependent Single Server Queue with Balking Al-Seedy, R.O.; El-Sherbiny, A.A.; El-Shehawy, S.A.; Ammar, S.I. 3. Relating time- and customer- averages for queues using “forward” coupling from the past Peköz, Erol A.; Ross, Sheldon

31 10 – Sessions

4. A Time-Dependent View of ASTA and Little’s Law Fralix, B.H.; Serfozo, R.F.; Riaño, G.

Session MC09 - Inventory 2, Chair: Yao in CZ 16, see page 82

1. An EOQ model with state dependence, random yield, and static pricing Baron, Opher; Berman, Oded; Perry, David

2. Periodic review inventory models Bijvank, M.

3. Asymptotically Optimal Solutions to a Joint Replenishment-Transshipment Model Yao, D.D.; Xiao, Yongbo

4. Study on inventory management with existence of electronic markets Xie, Ming; Yin, Wenjun; Zhang, Bin; Bai, X.X.; Dong, Jin

Tuesday 8:30am - 10:00am

Session TA01 - Financial Engineering 3, Chair: Kou in CZ 4, see page 84

1. Pricing, hedging and calibrating credit from the top down Giesecke, Kay

2. Malliavin Greeks without Malliavin calculus Chen, Nan; Glasserman, Paul

3. To be determined Cont, R.

4. Pricing Asian option via double Laplace transform Cai, Ning; Kou, S.G.

Session TA02 - Stochastic Geometry, Chair: Van Lieshout in CZ 5, see page 85

1. Wiener sausage and sensor networks Spodarev, E.; Cerný, R.; Funken, S.; Meschenmoser, D.; Rataj, J.; Schmidt, V.

2. Maximum likelihood estimation for random sequential adsorption Van Lieshout, M.N.M.

3. Random field model for fingerprint features Thönnes, Elke

4. Opportunistic Routing on Signal-to-Interference-and-Noise Ratio Graphs Baccelli, F.; Blaszczyszyn, B.; Mühlethaler, P.

Session TA03 - Rare Event Simulation 2, Chair: Juneja in CZ 10, see page 86

1. Alternative efficiency proof and interpretations for a recent state-dependent importance sampling scheme De Boer, P.T.; Scheinhardt, W.R.W.

32 2. Importance sampling for heavy-tailed random walk Bassamboo, A.; Juneja, Sandeep; Zeevi, Assaf

3. A Reduced-Form Model With Short-Time Contagion For Credit Risk Blanchet, Jose; Zhan, Li

4. Minimizing Tail Probabilities: Asymptotic Analysis and Efficient Simulation Optimization Juneja, Sandeep; Kalra, Himanshu

Session TA04 - Scheduling and Stability, Chair: Hasenbein in CZ 11, see page 87

1. Fluid and Diffusion Limits of a Multiclass Overloaded Abandonment Queue under Priority Schedul- ing Jennings, O.; Reed, J.E.

2. Stability of Non-random Critically Loaded Networks under MaxWeight Scheduling Andrews, D.M.; Jung, K.; Stolyar, A.L.

3. Undecidability results in the theory of queueing networks and Skorokhod problem Bertsimas, D.; Gamarnik, D.; Katz, D.

4. Networks with Parameter Uncertainty: Stochastic Fluid Models with Recourse Hasenbein, John J.; Buke, Burak; Morton, David P.

Session TA05 - Ad Hoc Networks, Chair: Van den Berg in CZ 12, see page 88

1. Efficient Resource Control for the EDCA and HCCA Mechanisms in Multi-Rate IEEE 802.11e Net- works Siris, Vasilios A.; Courcoubetis, Costas

2. Performance Modeling of a Bottleneck Node in an IEEE 802.11 Ad-hoc Network Roijers, Frank; Van den Berg, J.L.; Mandjes, M.R.H.

3. Collaboration and incentives in wireless ad-hoc networks Gibbens, R.J.; Key, P.B.

4. A fluid model for a relay node in an ad-hoc network: the case of heavy-tailed input Bekker, R.; Mandjes, M.R.H.

Session TA06 - Call Centers 2, Chair: Koole in CZ 13, see page 90

1. Shift scheduling for heterogeneous contact centers via mixed-integer programming Helber, S.; Henken, K.

2. A real-time routing policy for call centers Pot, S.A.

3. The Hayward Approximation with Queueing Systems Ak¸sin,Z.; Van den Schrieck, Jean-Christophe

4. Staffing Call Centers with Differentiated Levels of Service: Approximate Solutions via Constraint Dualization Bassamboo, A.; Zeevi, Assaf

33 10 – Sessions

Session TA07 - Statistics 1, Chair: Segers in CZ 14, see page 91

1. Martingale approach on the average run lengths of Exponentially EWMA control charts Areepong, Y.

2. Comparative study of change detection properties of Poisson EWMA control procedure Sukparungsee, S.; Novikov, A.

3. Benchmarking of Fitting Algorithms for Continuous Phase-Type Distributions Pérez, J.F.; Riaño, G.

4. Regularly varying multivariate time series Basrak, Bojan; Segers, Johan

Session TA08 - Polling Models, Chair: Winands in CZ 15, see page 92

1. Polling systems with parameter renewals MacPhee, I.

2. M/G/∞ polling systems with random visit times Vlasiou, Maria; Yechiali, Uri

3. Polling Systems with Two-Phase Gated Service: Heavy Traffic Results for the Waiting-Time Distri- butions Van der Mei, R.D.; Resing, J.A.C.

4. Branching-type polling systems with large setups Winands, E.M.M.

Session TA09 - Switch Scheduling, Chair: Moallemi in CZ 16, see page 93

1. Switch scheduling via message passing Bayati, M.

2. How to avoid doing research “10 years ahead of time” Firoozshahian, Amin; Hosseinkeh, Vahideh; Goel, Ashish ; Prabhakar, B.

3. Packet Election For Fair Scheduling Jagabathula, S.; Doshi, V.; Shah, D.

4. Approximate Dynamic Programming for Switch Scheduling Moallemi, Ciamac C.; Kumar, Sunil; Van Roy, Benjamin

Tuesday 10:30am - 12:00pm

Session TB01 - Finance and Ruin, Chair: Collamore in CZ 4, see page 95

1. A Dirichlet bridge sampling of the variance gamma process: Pricing path-dependent options Ballotta, Laura; Dimitrova, Dimitrina; Kaishev, Vladimir

2. Applications of series representations for Lévy processes Michna, Z.

3. Stochastic recurrence equations and ruin in a general Markovian environment Collamore, J.F.; Nyrhinen, H.

34 4. Managing Inventory of Items with Replacement Warranty Huang, Wei; Kulkarni, Vidyadhar; Swaminathan, Jayashankar M.

Session TB02 - Random Partitions and Partition-valued Processes, Chair: Gnedin in CZ 5, see page 96

1. Small time asymptotics of Lambda-coalescents Berestycki, J.; Berestycki, N.; Limic, V.

2. Gaussian fluctuations for Plancherel partitions Bogachev, L.

3. Ancestral processes in population genetics - exchangeable coalescents Möhle, Martin

4. Random partitions and the site frequency spectrum in population genetics Berestycki, N.; Berestycki, J.; Schweinsberg, Jason

Session TB03 - Gradient Estimation, Chair: Heidergott in CZ 10, see page 97

1. Policy Gradient Methods in Machine Learning Peters, Jan; Theodorou, Evangelos; Schaal, Stefan

2. Strong Properties of Weak Derivatives Leahu, Haralambie; Heidergott, B.; Hordijk, A.

3. Gradient Estimation for a Multi-Component Maintenance System Using Measure-Valued Differen- tiation Yuan, Taoying; Heidergott, B.

4. Measure-Valued Differentiation for Random Horizon Experiments Heidergott, B.; Hordijk, A.

Session TB04 - Asymptotic Analysis 2, Chair: Xia in CZ 11, see page 99

1. Are end-to-end acknowledgements causing power law delays in large multi-hop networks? Jelenkovic,´ P.R.; Tan, J.

2. Stochastic ordering for regulated flows and the Better-than-Poisson Property Guillemin, F.; Mazumdar, R.R.; Rosenberg, C.P.; Ying, Yu

3. Sharpness: a sufficient and necessary condition for throughput scalability Chaintreau, A.

4. Scalability of Fork/Join Queueing Networks with Blocking Xia, Cathy H.; Liu, Zhen; Towsley, Don; Lelarge, Marc

Session TB05 - Randomized Algorithms and Resource Control Schemes, Chair: Borst in CZ 12, see page 100

1. Optimal peer-to-peer broadcasting schemes Massoulié, Laurent; Twigg, Andy

35 10 – Sessions

2. Mean field limits of particle systems in varying environments, Application to modeling random multi-access algorithms Bordenave, C.; McDonald, D.R.; Proutière, A.

3. Metastability of CDMA cellular systems Antunes, N.; Fricker, C.; Robert, Ph.; Tibi, D.

4. Ranking and suggesting tags in collaborative tagging applications Vojnovic, Milan

Session TB06 - Many-server Models 2, Chair: Van Leeuwaarden / Zwart in CZ 13, see page 102

1. Corrected diffusion approximations for many-server queues I Janssen, A.J.E.M.; Van Leeuwaarden, Johan S.H.; Zwart, Bert

2. Corrected diffusion approximations for many-server queues II Janssen, A.J.E.M.; Van Leeuwaarden, Johan S.H.; Zwart, Bert

3. A new view on the G/GI/N queue in Halfin-Whitt regime Reed, J.E.

4. Moment Equivalences Between Multiserver and Single Server Queues Scheller-Wolf, A.; Vesilo, R.A.

Session TB07 - Stochastic Control and PDE, Chair: Atar in CZ 14, see page 103

1. Optimal Stopping and Free Boundary Characterizations for some Brownian Control Problems Budhiraja, A.; Ross, Kevin J.

2. A control-theoretic analysis of fast-sweeping methods for static Hamilton-Jacobi equations Dupuis, Paul

3. HJB equations, no arbitrage, and generalized Brownian networks Atar, R.; Budhiraja, A.; Williams, R.J.

4. A Representation for the Characteristic Functional of Banach Valued Stable Random Measures Mahmoodi, S.; Soltani, A.R.

Session TB08 - Fluid Models, Chair: Scheinhardt in CZ 15, see page 104

1. Analysis of a resource sharing policy for computer systems using fluid queues Gautam, N.; Mahabhashyam, S.R.

2. A feedback fluid queue with two congestion control thresholds Malhotra, R.; Mandjes, M.R.H.; Scheinhardt, W.R.W.; Van den Berg, J.L.

3. On shot noise fluid queues and infinite-server queues with batch arrivals: time-dependent behavior Scheinhardt, W.R.W.; Boucherie, R.J.; De Graaf, W.F.

4. Tandem Queue with Server Slow-down Miretskiy, D.I.; Scheinhardt, W.R.W.; Mandjes, M.R.H.

36 Session TB09 - Supply Chains, Chair: Ettl / De Kok in CZ 16, see page 106

1. Centralized and decentralized vendor-managed inventory optimization under different levels of in- formation sharing Ettl, M.; Lu, Y.; Squillante, M.S.

2. Optimal Policies for Joint Replenishment and Substitution Xu, H.; Yao, D.D.; Zheng, S.

3. Benefits and risks of price protection in the computer industry Ettl, M.; Huang, P.; Kapuscinski, R.; Sourirajan, K.

4. Period Order Quantities in Multi-Item Multi-Echelon Inventory Systems De Kok, Ton

Tuesday 2:45pm - 4:15pm

Session TC01 - Statistics 2, Chair: Kuhnt in CZ 4, see page 107

1. Statistical Analysis of Spatial Patterns of Customer Behaviors Bai, X.X.; Yin, Wenjun; Dong, Jin

2. On the performance of estimators of parameters in autoregressive model under asymmetric loss Roychowdhury, Soma; Bhattacharya, Debasis

3. On nonparametric inference for a class of queuing systems Pawlak, M.

4. An application of algebraic statistics to Bowker’s test for symmetry Kuhnt, S.; Krampe, A.

Session TC02 - Anomalous Diffusion, Chair: Metzler in CZ 5, see page 108

1. Anomalous Subdiffusion Revisited Klafter, Joseph

2. Anomalous Diffusion Leads to Weak Ergodicity Breaking Barkai, E.

3. Mesoscopic approach to reactions between subdiffusing particles Sokolov, Igor M.

4. Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation Weron, Aleksander; Magdziarz, M.

Session TC03 - Cross-entropy, Chair: Kroese in CZ 10, see page 109

1. Parametric Minimum Cross-Entropy Method Rubinstein, R.Y.; Dolgin, A.; Kroese, D.P.; Glynn, P.W.

2. A Nonparametric Extension of the Cross-Entropy method Botev, Z.I.; Taimre, T.; Kroese, D.P.

3. A Generalized Cross-Entropy Method: Discrete State Spaces Taimre, T.; Botev, Z.I.; Kroese, D.P.

37 10 – Sessions

4. Adaptive importance sampling algorithms for rare-event simulations of the infinite-server queue. Ridder, A.

Session TC04 - Limit Theorems for Queueing Systems with Measure-valued State Descriptors, Chair: Puha in CZ 11, see page 111

1. Limit Theorems for Many-Server Queues Ramanan, K. 2. Heavy Traffic Analysis for EDF Queues with Reneging Kruk, Lukasz; Lehoczky, J.; Ramanan, K.; Shreve, S. 3. Central Limit Theorems for a large queuing network with resource pooling Graham, C. 4. The Fluid Limit of a Shortest Remaining Processing Time Queue Down, Douglas G.; Gromoll, H.C.; Puha, A.L.

Session TC05 - Communication Systems 2, Chair: Choi in CZ 12, see page 112

1. Performance analysis of small non-uniform packet switches Beekhuizen, P.; Resing, J.A.C. 2. Performance analysis of sleep mode and idle mode for best effort traffic in IEEE 802.16e Lee, Yong Hyun; Hwang, Eunju; Kim, Kyung Jae; Son, Jung Je; Choi, Bong Dae 3. Performance Analysis of Power Saving Class of Type 1 for both Downlink and Uplink Traffics in IEEE 802.16e Baek, Sangkyu; Son, Jung Je; Lee, Seon Mi; Choi, Bong Dae 4. A New Power Saving Mechanism with Periodic Traffic Indications in the IEEE 802.16e Hwang, Eunju; Kim, Kyung Jae; Son, Jung Je; Choi, Bong Dae

Session TC06 - Matrix-Analytic Methods, Chair: Squillante / Taylor in CZ 13, see page 114

1. Bounds and Decay Rate for a QBD Process with Countably Many Background States Li, H.; He, Q.-M.; Zhao, Y.Q. 2. Algorithmic approach to the extinction probability of branching processes Hautphenne, S.; Latouche, Guy; Remiche, M.-A. 3. Physical interpretations of algorithms for stochastic fluid flows Bean, Nigel; O’Reilly, Małgorzata; Taylor, Peter 4. Invariant measures for transient fluid queues Van Lierde, Sarah; Da Silva, Ana; Latouche, Guy

Session TC07 - MDP 2, Chair: Örmeci in CZ 14, see page 115

1. A Semi Markov Decision Approach for Train Conflict Resolution Al-Ibrahim, Assil; Van der Wal, Jan 2. Exact finite approximations of average-cost countable Markov Decision Processes Leizarowitz, Arie; Shwartz, Adam

38 3. A Controlled Markov Process Model for a Benchmark Reentrant Manufacturing Line with Infinite and Finite Capacity Buffers Ramírez-Hernández, José A.; Fernandez, Emmanuel

4. Effects of System Parameters on the Optimal Policy Structure in a Class of Queueing Control Prob- lems Ormeci, E.L.; Cil, E.B.; Karaesmen, F.

Session TC08 - Queueing Models 2, Chair: Gómez-Corral in CZ 15, see page 116

1. Bad luck when joining the shortest queue Blanc, Hans

2. On the correlation structure of Gaussian queues Es-Saghouani, A.; Mandjes, M.R.H.

3. Negative Queue Length:Queues for Quantum Information Toyoizumi, Hiroshi

4. On a batch Markovian arrival process subject to renewal catastrophes Gómez-Corral, A.

Session TC09 - World Wide Web, Chair: Litvak in CZ 16, see page 117

1. PageRank of Scale-Free Growing Networks Avrachenkov, K.; Lebedev, D.

2. Distances in Random Graphs Van der Hofstad, R.

3. Stochastic equations behind Web ranking Volkovich, Y.; Donato, D.; Litvak, Nelly; Scheinhardt, W.R.W.

4. A fair distribution of probability mass on the bow-tie graphs Avrachenkov, K.; Litvak, Nelly; Pham, Kim Son

Tuesday 4:45pm - 6:15pm

Session TD01 - Lévy Processes and Finance, Chair: Rieder / Bäuerle in CZ 4, see page 119

1. Malliavin Calculus for Levy Processes and Applications in Finance Petrou, Evangelia

2. Extremal behavior of stochastic integrals driven by regularly varying Lévy processes Hult, H.; Lindskog, F.

3. Dependence properties and comparison results for Lévy processes Bäuerle, N.; Müller, A.; Blatter, A.

39 10 – Sessions

Session TD02 - Spatial Models 1, Chair: Foss / Stolyar in CZ 5, see page 120

1. Large deviations of the interference in a wireless communication model Ganesh, A.J.; Torrisi, G.L.

2. M/D/1/1 loss system with interference and applications to transmit-only sensor networks Blaszczyszyn, B.; Radunovic,´ B.

3. Local and global properties of random directed trees Roy, Rahul

4. Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models Lelarge, Marc; Aldous, David; Bordenave, C.

Session TD03 - Asymptotics, Chair: Blanchet / Mikosch in CZ 10, see page 121

1. Towards Efficient Simulation of Long Delays for Heavy-tailed Multiserver Queues Blanchet, Jose; Glynn, P.W.

2. Large deviations for point processes based on stationary sequences with heavy tails Hult, H.; Samorodnitsky, G.

3. Rare event simulation methodology for perpetuities Blanchet, Jose; Zwart, Bert

4. Boundary behavior and stability for multidimensional reflected diffusions with state-dependent jumps in the positive orthant Mazumdar, R.R.; Guillemin, F.; Piera, F.

Session TD04 - Large Deviations 1, Chair: Majewski in CZ 11, see page 122

1. Large deviations of multitype queues Dabrowski, A.; Lee, J.; McDonald, D.R.

2. Limiting distribution of the phase when level passes a threshold Khanchi, Aziz; McDonald, D.R.

3. Large Deviations of Generalized Jackson Networks Meiner, Silke

4. Functional asymptotics of a simple queueing network with short- and long-range dependent input Majewski, Kurt

Session TD05 - Computer-communications, Chair: Hwang in CZ 12, see page 123

1. Asymptotic results in some random infinite urn models Robert, Ph.; Simatos, F.

2. Connectivity probability of one-dimensional ad hoc wireless networks with minimum hop path Antunes, N.; Jacinto, Gonçalo; Pacheco, A.

3. Design and Analysis of a Packet Scheduler Exploiting the Multiuser Diversity for Performance En- hancement Hwang, Gang Uk; Ishizaki, Fumio

40 Session TD06 - Applied Probability, Chair: Hagwood in CZ 13, see page 124

1. Stochastic modeling of a dark matter experiment Coakley, Kevin 2. Stochastic Modeling in Gene Expression Measurements: Accounting for Both Background Error and Instrument Saturation Lu, Z.Q. John 3. Stochastic Modeling of a Carbon Nanotube in an Electrical Field Hagwood, Charles 4. Allocation Models and heuristics for the outsourcing of repairs for a dynamic warranty population Ding, L.; Glazebrook, K.D.; Kirkbride, C.

Session TD07 - Optimal Stopping 1, Chair: Gnedin in CZ 14, see page 125

1. Maximizing the expected duration time related to Poisson arrival Kurushima, A.; Ano, K. 2. A linear programming analysis of finite horizon optimal stopping problems Helmes, K. 3. Optimal stopping with a rank-dependent loss Gnedin, A.V. 4. Optimal Stopping of ”Seasonal” Observations Presman, E.L.; Sonin, Isaac

Session TD08 - Queueing in Telecom, Chair: Tijms in CZ 15, see page 127

1. A spectral method for a BMAP/D/c queue with synchronous servers Nishimura, S.; Yatomi, N. 2. Differential Traffic Theory Garcia, J.M.; Brun, O. 3. Moments of the duration of busy periods in oscillating queueing systems Pacheco, A.; Ribeiro, H. 4. A quick approximation for the state probabilities and waiting-time probabilities in the M/D/1 queue Tijms, H.C.

Session TD09 - Engineering, Chair: Günalay in CZ 16, see page 128

1. Analysis of a Probabilistic Key Management Scheme for Vehicular Communications Van den Berg, E. 2. Comparison of the effect of various forecasting methods on bullwhip effect in supply chain Farahani, R.Z.; Najafi, M.; Asgari, N. 3. Using Conditional Probabilities for Automatic New Topic Identification of Search Engine Transac- tion Logs Ozmutlu, Seda; Ozmutlu, Huseyin C.; Buyuk, Buket 4. A queueing analysis of a hybrid manufacturing system with new and refurbished items Günalay, Y.; Gupta, D.

41 10 – Sessions

Wednesday 8:30am - 10:00am

Session WA01 - Queues and Lévy Processes, Chair: Kella in CZ 4, see page 129

1. The M/G/1 queue with quasi-restricted accessibility Perry, David

2. Lévy correlation cascades Eliazar, Iddo

3. Superposition of renewal processes and an application to multi-server queues Stadje, Wolfgang

4. Lévy driven fluid queues with negatively correlated service interruptions and busy periods Kella, Offer

Session WA02 - Random Graphs, Chair: Norros in CZ 5, see page 130

1. Two diameter results in the configuration model Hooghiemstra, Gerard

2. A local limit theorem for the critical random graph Müller, T.; Van der Hofstad, R.; Kager, W.

3. Percolation on the 2D Hamming graph: The supercritical phase Luczak, M.J.; Van der Hofstad, R.

4. Robustness against core losses in a power-law random graph Norros, I.; Reittu, H.

Session WA03 - Simulation 2, Chair: Grassmann in CZ 10, see page 132

1. Zero-Variance Importance Sampling Estimators for Markov Process Expectations Awad, H.P.; Glynn, P.W.; Rubinstein, R.Y.

2. Fluid Limit for Hybrid MCMC samplers Fort, G.; Moulines, E.; Priouret, P.

3. Applied Probability Contribution to the Burn-in Phase in Equilibrium Simulation Grassmann, W.K.; Sethi, S.

4. Ordering of time correlations in stationary Markov processes and queueing networks Szekli, R.; Daduna, H.

Session WA04 - Networks, Chair: Zazanis in CZ 11, see page 133

1. Functional diffusion approximation of the queue with an infinity of servers Decreusefond, L.; Moyal, P.

2. Diffusion approximations for reinsurance problems Zazanis, Michael; Yannacopoulos, A.N.; Papanicolaou, V.; Frangos, E.N.

3. Active-Idleness and the Multiclass Queueing Networks Optimization using Infinitesimal Perturba- tion Analysis Orfão, N.M.; Bispo, C.F.

42 4. On Stochastic Network Calculus Jiang, X.

Session WA05 - Internet, Chair: Fricker in CZ 12, see page 135

1. Transient analysis and asymptotic of the mean of the first hitting time for a TCP window size process Löpker, Andreas H.; Van Leeuwaarden, Johan S.H.

2. Multiclass G/M/1 queueing system with self-similar input and non-preemptive priority Caglar,˜ M.; Iftikhar, M.; Landfeldt, B.; Singh, T.

3. Sampling Internet traffic Antunes, N.; Chabchoub, Y.; Fricker, C.; Guillemin, F.; Robert, Ph.

Session WA06 - Infinite Multidimensional Markov Chains, Chair: Squillante / Tay- lor in CZ 13, see page 136

1. Geometric decay in a QBD process with countable background states with applications to a join-the- shortest-queue model Li, H.; Miyazawa, M.; Zhao, Y.Q.

2. Stability Analysis of a General Class of Markov Chains Gamarnik, D.; Squillante, M.S.

3. Some interesting decay rate behaviour in quasi-birth-and-death processes with infinite phase spaces Motyer, Allan; Taylor, Peter

4. Explicit solutions and decay rates for a general class of Markov chains Van Leeuwaarden, Johan S.H.; Squillante, M.S.; Winands, E.M.M.

Session WA07 - Optimal Stopping 2, Chair: Sonin in CZ 14, see page 137

1. Fiducial approach in the invariant problem of optimal choice Belenky, Vitali Z.

2. Bandwagon Random Walks Quinn, J.

3. A Generalized Gittins Index for Markov Chain and its Recursive Calculation Sonin, Isaac

Session WA08 - Flexible Servers, Chair: Down in CZ 15, see page 138

1. Tandem Lines with Synergistic Servers Andradóttir, Sigrún; Ayhan, Hayriye; Down, Douglas G.

2. A call center model with upgrades Lewis, Mark E.; Down, Douglas G.

3. Resource allocation in grid computing Koole, G.; Righter, R.

4. Linear Programming Based Affinity Scheduling for Heterogeneous Computing Systems Down, Douglas G.; Al-Azzoni, Issam

43 10 – Sessions

Session WA09 - Inventory 3, Chair: Van Houtum in CZ 16, see page 139

1. Approximate evaluation of order fill rates for an inventory system of service tools Vliegen, I.M.H.; Van Houtum, G.J.

2. Exact and Approximate Results for the Kit Management Problem Gullu, R.; Koksalan, M.

3. Valuation of downstream liquefied natural gas storage Scheller-Wolf, A.; Wang, Xiaofeng (Mulan); Kekre, Sunder; Secomandi, Nicola

4. Analysis of an inventory model with lateral transshipment Kranenburg, A.A.; Van Houtum, G.J.

Wednesday 10:30am - 12:00pm

Session WB01 - Last Minute Talks, Chair: Boxma in CZ 4, see page 141

1. New Algorithms for Laplace Transform Inversion Den Iseger, Peter

Session WB02 - Self-interacting Stochastic Processes, Chair: Van der Hofstad in CZ 5, see page 141

1. An expansion for self-interacting random walks Holmes, M.

2. Parabolic Anderson model: Localisation of mass in random media König, W.; Mörters, P.; Sidorova, N.

3. Excited to the center Kozma, G.

4. Sampling for the maximum of Brownian motion Van Leeuwaarden, Johan S.H.

Session WB03 - Stochastic Programming and Optimization 2, Chair: Veatch in CZ 10, see page 142

1. Efficient Allocation Under Incomplete Information Burnetas, Apostolos N.

2. A problem of optimal sequential pricing Hodge, D.J.

3. Cutting speed adjustment and tool magazine capacity allocation using stochastic dynamic program- ming Lamond, B.F.

4. Approximation, structure and performance for average cost queueing network control Veatch, Michael

44 Session WB04 - Fluid Limits and Mean Fields, Chair: D’Auria in CZ 11, see page 143

1. Mean field analysis of discrete-time closed queueing networks with geometric service times. Adan, I.J.B.F.; Beekhuizen, P.; D’Auria, B.

2. Spontaneous resonances and the onset of collective behavior in queuing networks Vladimirov, A.A.; Rybko, A.N.; Shlosman, S.B.

3. Conic Instability of fluid limit models Kim, Bara

Session WB05 - TCP, Chair: Baccelli in CZ 12, see page 144

1. Iterated function systems and ergodicity of AIMD networks King, C.; Shorten, R.; Wirth, F.; Leith, D.

2. Queueing Theory for TCP Wischik, D.

3. A stochastic model of Split TCP Baccelli, F.; Carofiglio, G.; Foss, S.

4. The Equations of TCP Baccelli, F.; Kim, Ki Baek; McDonald, D.R.

Session WB06 - Call Centers 3, Chair: Koole in CZ 13, see page 146

1. Modeling arrivals Chevalier, Philippe; Van den Schrieck, Jean-Christophe

2. An approximation method for evaluating the performances of a skill-based routing call center model Jouinni, Oualid; Salah Aguir, Mohamed; Dallery, Yves

3. Design and Capacity Planning in Multi-Skill Call Centers Ormeci, E.L.; Cakan, N.; Ak¸sin,Z.; Karaesmen, F.

4. Dynamic call blending in the presence of time-varying arrivals Balint, A.; Bhulai, S.

Session WB07 - Dynamic Control for Service Management, Chair: Hampshire in CZ 14, see page 147

1. A Modified Fluid Model for the Optimal Design of a Dynamic Call Center Massey, Bill

2. Inventory rationing for a system with a lost-sales and a backordering customer class Enders, P.; Adan, I.J.B.F.; Van Houtum, G.J.

3. Dynamic Pricing and Capacity Planning of e-Services under Demand Uncertainty Xiu, Cathy

4. Efficient Frontier Dynamic Pricing for On-Demand Music and Movie Services Hampshire, Robert C.

45 10 – Sessions

Session WB08 - Queueing Models 3, Chair: Brandt in CZ 15, see page 149

1. Necessary and Sufficient Conditions for Finite First Moment of the Workload Components in FIFO Multiserver Queues Scheller-Wolf, A.; Vesilo, R.A.

2. Comparison on Queueing Systems with Generalized Processor Sharing Scheduling Kim, Sunggon; Lee, Ju Yong; Park, Wonyoung

3. Approximations for the M/GI/N + GI Type Call Center Balcıoglu,˜ Baris; Travani, Foad

4. Waiting times for M/M systems under generalized processor sharing Brandt, M.; Brandt, A.

Session WB09 - Combinatorial Structures, Chair: Gamarnik / Ramanan in CZ 16, see page 150

1. Probabilistic analysis of game tree evaluation Neininger, R.

2. On Temperate Regions in Random Groves Baryshnikov, Y.

3. The mixing advantage is less than 2 Hamza, Kais; Jagers, P.; Sudbury, A.; Tokarev, D.

4. Iterative Algorithm for Maximum Weight Independent Set Sanghavi, S.; Shah, D.; Willsky, A.

Wednesday 1:30pm - 3:00pm

Session WC01 - Risk and Insurance, Chair: Kolbe in CZ 4, see page 151

1. Optimal Insurance Demand and Investment in a Dynamic Mean-variance Framework Biffis, E.

2. Long-term survivor mixture models for credit scoring Mo, S.F.; Yau, K.K.W.

3. Stability and equilibria in incomplete markets Zitkovic,´ Gordan

4. Valuation and pricing of reverse mortgages Kolbe, Andreas; Zagst, Rudi

Session WC02 - Spatial Models 2, Chair: Foss / Stolyar in CZ 5, see page 152

1. Dynamics of pattern formation in multi-hop wireless networks using backoff-based MAC protocols Durvy, Mathilde; Thiran, Patrick

2. Optimal throughput allocation and stability in a spatial random-access model Stolyar, A.L.

46 3. A Random Multiple Access Protocol with Spatial Interactions Bordenave, C.; Foss, S.; Shneer, V.

4. Routing in Sensor Networks by Using Spatial Dynamics Ephremides, Anthony

Session WC03 - Perturbation and Discrete Events, Chair: Heidergott in CZ 10, see page 154

1. Law of Large Numbers for products of random matrices in the max-plus semiring Merlet, G.

2. Optimal short-term utilization bounds on multiclass networks Orfão, N.M.; Bispo, C.F.

3. Some new results on the performance optimization of queueing systems Xia, Li; Yin, Wenjun; Dong, Jin

4. Sensitivity analysis for Gaussian systems, a mathematical “urban legend” busted Heidergott, B.; Vázquez-Abad, F.; Volk-Makarewicz, W.

Session WC04 - Large Deviations 2, Chair: Ignatiouk-Robert in CZ 11, see page 155

1. Large Deviation and Fluctuation of Diffusion Processes with discontinuous drift Chiang, T.S.; Sheu, S.J.

2. Large deviations asymptotics of exit time and exit location for a class of Lagrangian systems Kovaleva, Agnessa

3. Martin boundary with large deviation technique for a random reflected random walks on a half-space Ignatiouk-Robert, Irina

4. Application of Tauberian theorem to the exponential decay of the tail probability of a random variable Nakagawa, K.

Session WC05 - Bandwidth Sharing Networks, Chair: Borst in CZ 12, see page 157

1. Bandwidth-Sharing Networks in Overload Egorova, R.; Borst, S.C.; Zwart, Bert

2. Stability of bandwidth sharing networks with monotone service rates Jonckheere, Matthieu

3. Importance sampling in rate-sharing networks Lieshout, P.; Mandjes, M.R.H.

4. Flow-level Stability of Data Networks with Non-convex and Time-varying Rate Regions Liu, J.; Proutière, A.; Yi, Y.; Chiang, M.; Poor, H.V.

Session WC06 - Biology, Chair: Spencer in CZ 13, see page 158

1. Determining Hidden Markov Models Efficacy in gene finding problem Kazemnejad, A.; Hajizadeh, E.; Mirjafari, K.

47 10 – Sessions

2. Count or control? Optimal monitoring for invasive species management McCarthy, M.A.; Moore, A.; Taylor, Peter 3. Using stochastic epidemic models to determine if emerging avian influenza can escape containment Spencer, S.E.F.; O’Neill, P.D.

Session WC07 - Games 2, Chair: Weber in CZ 14, see page 159

1. Symmetric Rendezvous Games Fan, J. 2. On two-stopping and stopping game problems Szajowski, K. 3. Symmetric rendezvous search Weber, R.R.

Session WC08 - Queueing Models 4, Chair: Vlasiou in CZ 15, see page 160

1. Analysis of a queueing system with synchronized services and setup times. Economou, A.; Kapodistria, S.

2.A Geo/G/1/∞ queueing system with a generalized N-policy and setup-closedown times Moreno, P. 3. Production inventory with service time and vacation to the server Krishnamoorthy, A.; Narayanan, Viswanath C. 4. A multi-station queue with dependent interarrival and service times Vlasiou, Maria

Session WC09 - Health Care, Chair: Ziya in CZ 16, see page 161

1. Optimal outpatient appointment scheduling Kaandorp, G.C.; Koole, G. 2. Nurse Staffing and Bed Capacity within Hospitals Units Jennings, O. 3. Managing New Patient Admissions in Multi-site Healthcare Facilities Tekin, E.

4. Identifying the Critical Patient: Finding the Balance between Undertriage and Overtriage Ziya, S.; Argon, N.T.

48 11 Abstracts

Monday 10:45am - 12:15pm

Session MA01 - Financial Engineering 1, Chair: Kou in CZ 4

Optimal Consumption and Portfolio Management with Non-Observable Real Prices

A. Bensoussan, University of Texas at Dallas, USA, [email protected] J. Keppo, University of Michigan, USA, [email protected] S.P. Sethi, University of Texas at Dallas, USA, [email protected]

We consider the classical optimal consumption and portfolio investment problem subject to a random infla- tion in the consumption good prices over time. Both the cases when the investor observes the consumption good prices and when he receives noisy observations on those are treated. We derive the optimal policy and show that a modified Mutual Fund Theorem consisting of three funds holds in both cases. The formulas for the compositions of the funds in the two cases are the same, but in general their actual compositions differ because in the noisy observation case the consumption basket is a random variable. However, when the investor has a CRRA utility, then his optimal portfolio strategy is the same in the two cases. The additional risk due to the non observability decreases consumption and increases saving.

Robustness and sensitivity analysis of risk measurement procedures

R. Cont, IEOR Department – Columbia University, United States, [email protected] R. Deguest, Centre de Mathématiques Appliquées – Ecole Polytechnique, France, [email protected] G. Scandolo, Dipartimento di Matematica per le Decisioni – Università di Firenze, France, giacomo.scandolo@unifi.it

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the port- folio from available observations and computing a risk measure of this distribution which summarizes the risk of the portfolio. These two steps are usually considered and studied separately. We define the notion of risk measurement procedure, as a two–step procedure which yields a correspondence between a couple (DN , X), where X is a payoff and DN a data set on which the estimation is based, and a risk estimate R(DN , X) for X. This quantity is supposed to estimate the “abstract” risk measure ρ(P, X) where X is payoff and P is the “true” probability measure from which the data set is generated. It allows us to study some natural requirements of the risk measurement procedure: coherence, robustness and sensitivity. We state some theoretical results which illustrate a conflict between the coherence of a risk measurement pro- cedure and its robustness. We complement this abstract result by computing measures of sensitivity, which allow to quantify the robustness of various risk measures with respect to the data set used to compute them.

49 11 – Abstracts

In particular, we show that the same “risk measure” may exhibit quite different sensitivities depending on the estimation procedure used.

What Is a Good Risk Measure: Bridging the Gaps between Robustness, Subaddi- tivity, and Insurance Risk Measures

C.C. Heyde, Columbia University, The United States, [email protected] S.G. Kou, Columbia University, The United States, [email protected] X.H. Peng, Columbia University, The United States, [email protected]

Two main axiomatically based risk measures are the coherent risk measure, which assumes subadditivity for random variables, and the insurance risk measure, which assumes additivity for comonotonic random variables. We propose a new, data based, risk measure, called natural risk statistic, that is characterized by a new set of axioms. The new axioms only require subadditivity for comonotonic random variables, which is consistent with the prospect theory in psychology. Comparing to two previous measures, the natural risk statistic includes the tail conditional median which is more robust than the tail conditional expectation suggested by the coherent risk measure; and, unlike insurance risk measures, the natural risk statistics can also incorporate scenario analysis. The natural risk statistic includes the VaR as a special case and therefore shows that VaR, though simple, is not irrational.

Dynamic importance sampling for credit baskets

S. Weber, ORIE, Cornell University, 279 Rhodes Hall, Ithaca, NY 14853, [email protected] M. Liese, ETH Zurich, Switzerland, [email protected]

We discuss dynamic importance sampling techniques which improve the efficiency of Monte Carlo estima- tors of the prices of credit derivatives like credit basket swaps and CDOs.

Session MA02 - Random Fields, Chair: Kaj in CZ 5

Operator Scaling Stable Random Fields: Theory and Simulation

H.P. Scheffler, University of Siegen, Germany, scheffl[email protected] H. Biermé, MAP5 Université René Descartes, France, [email protected] M.M. Meerschaert, Michigan State University, USA, [email protected]

A random field is a family of random variables indexed by points in an Euclidian parameter space. Classical examples such as the fractional Brownian field have stationary increments and are self-similar with Hurst index H. Moreover they are isotropic, that is invariant under rotations of the underlying parameter space. In many applications, for example the modelling of fractured rock however, the field should be anisotropic and should have different self-similarity indices in different directions. Examples of such fields are the so-called operator scaling random fields. Two different and usually not equivalent representations of such fields will be presented as well as an efficient simulation method for some important special cases.

Random fields generated from multifractional pseudodifferential operators: Pseu- dostable random fields

M.D. Ruiz-Medina, Granada University, Granada, Spain, [email protected] J.M. Angulo, Granada University, Granada, Spain, [email protected] V.V. Anh, Queensland University of Technology, Brisbane, Australia, [email protected]

50 MA02

In the Gaussian case, random fields defined by multifractional elliptic and parabolic equations are charac- terized in terms of Reproducing Kernel Hilbert Space (RKHS), and multifractional Sobolev space theory (Ruiz-Medina et al., 2004b, and Ruiz-Medina et al., 2007). Their local Hölder exponent is a function depending on space, and on space and time, respectively. On the other hand, under certain conditions, mul- tifractional pseudodifferential operators also generate Feller semigroups. Markov processes with transition probability densities having multifractional local Hölder exponent can then be introduced in this frame- work (see Jacob, 2005). An interesting special case is the so called pseudostable processes (Lévy-type processes), which can be generated from the parametrix of the Riesz potential of variable order (see Zahle, 2004, for the fixed-order case). We also consider the formulation of these models on non-regular compact domains (Ruiz-Medina et al., 2004a). In particular, heterogeneous domains with variable local dimension are studied in Ruiz-Medina et al. (2007). The multifractal domain case is introduced as a limiting case when regularity assumptions on the multifractional exponent defining the local dimension of the domain, the local Hölder exponent of the associated d(·)-measure, are relaxed .

Acknowledgements. This work has been supported in part by projects MTM2005-08597 of the DGI and P05-FQM-00990 of the Andalou- sian CICE, Spain. References Jacob, N. (2005). Pseudodifferential Operators and Markov Processes, Vol. III. Imperial College Press. Ruiz-Medina, M.D., Angulo, J.M. and Anh, V.V. (2007). Multifractality in Space-Time Statistical Models. Stochastic Environmental Research and Risk Assessment (to appear). Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M. (2004a). Fractal random fields on domains with fractal boundary. Infinite Dimensional Analysis, Quantum Probability and Related Topics 7, 395-417. Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M. (2004b). Fractional generalized random fields of variable order. Stochastic Analysis and Applications 22, 775-800. Ruiz-Medina, M.D., Anh, V.V. and Angulo, J.M. (2007). Random fields with multifractional regularity or- der on multifractals. Infinite Dimensional Analysis, Quantum Probability and Related Topics (submitted). Zahle, M. (2004). Riesz potentials and Liouville operators on fractals. Potential Analysis 21, 193-208.

Self-similar random fields as scaling limits of Poisson germ-grain models

Ingemar Kaj, Department of Mathematics, Uppsala University, Sweden, [email protected] Anne Estrade Hermine Biermé

We study limiting random systems that arise by aggregation of spherical grains which are uniformly scat- tered according to a Poisson point process in d-dimensional space. The grains have random radius, in- dependent and identically distributed, with a distribution which is assumed to have a power law behavior either in zero or at infinity. The resulting configurations of mass suitably centered and normalized exhibit limit distributions under scaling, which are conveniently described in a random fields setting. In this man- ner we construct Gaussian self-similar random fields of index H, for arbitrary noninteger H > −d/2. We obtain also non-Gaussian random fields with interesting properties, in particular a model of the type "fractional Poisson motion". References: I.Kaj, L. Leskelä, I. Norros, V. Schmidt, Scaling limits for random fields with long-range dependence. Ann. Probab. 35:2 (2007), 528-550. H. Biermé, A. Estrade, I. Kaj, About scaling behavior of random balls models. 6th Int. Conf. on Stereology, Spatial Statistics and Stochastic Geometry, Prague, June 2006.

51 11 – Abstracts

Characterization of Banach Valued Stable Random Measures

S. Mahmoodi, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran. [email protected] A.R. Soltani, Department of Statistics & Operations Research, Faculty of Science, Kuwait University, State of Kuwait. [email protected]

A Banach space (X: real separable) valued symmetric α stable random measure 8 with independent increments on a measurable space (F, F ) is considered. A pair (k, µ), called here a control pair, for + which k : X × F → R , µ a positive measure on (F, F ), is introduced. It is proved that the law of 8 is governed by a control pair; and every control pair will induce such 8. Moreover k is unique for a given µ. Our derivations are based on the Generalized Bochner Theorem and the Radon- Nikodym Theorem for vector measures.

Session MA03 - Rare Event Simulation 1, Chair: Dupuis in CZ 10

Fast Simulation of Brownian Motion Avoiding Poissonian Obstacles

Jose Blanchet, [email protected] Paul Dupuis, [email protected]

We are interested in developing efficient rare-event simulation methodology for the probability that, for long time, Brownian motion avoids a cloud of obstacles that are distributed according to an homogeneous Poisson process. This problem, motivated from the physics literature, is related to the development of long time asymptotics for the Laplace transform of the volume of the Wiener sausage. We investigate several simulation strategies that are suggested by celebrated large deviations results on the volume of the Wiener sausage and discuss their theoretical performance.

Fast pricing of basket default swaps

Zhiyong Chen, Bear Stearns, New York Paul Glasserman, Columbia University, New York, [email protected]

A basket default swap is a derivative security tied to an underlying basket of assets subject to credit risk. The value of the contract depends on the joint distribution of the default times of the underlying assets. Valuing a basket default swap often entails simulation of these default times. For baskets of high-quality credits and for swaps that require multiple defaults to trigger payment, pricing the swap is a rare-event simulation problem. Joshi and Kainth (2004) introduced an innovative importance sampling technique for this problem that forces a predetermined number of defaults to occur on each path. This paper analyzes, extends and improves the Joshi-Kainth algorithm. We show that, in its original form, the algorithm can actually increase variance; we present an alternative that is guaranteed to reduce variance, even when defaults are not rare. Along the way, we provide a rigorous underpinning in a setting sufficiently general to include both the original method and the version proposed here.

Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Ran- dom Variables

Paul Glasserman, Columbia University, [email protected] Sandeep Juneja, Tata Institute of Fundamental Research, India, [email protected]

52 MA04

Successful efficient rare event simulation typically involves using importance sampling tailored to a specific rare event. However, in applications one may be interested in simultaneous estimation of many probabilities or even an entire distribution. In this talk, we address this issue in a simple but fundamental setting. Specifically, we consider the problem of efficient estimation of the probabilities P(Sn ≥ na) for large n, for all a lying in an interval A, where Sn denotes the sum of n independent, identically distributed light-tailed random variables. Importance sampling based on exponential twisting is known to produce asymptotically efficient estimates when A reduces to a single point. We show, however, that this procedure fails to be asymptotically efficient throughout A when A contains more than one point. We analyze the best performance that can be achieved using a discrete mixture of exponentially twisted distributions and then present a method using a continuous mixture. We show that a continuous mixture of exponentially twisted probabilities and a discrete mixture with a sufficiently large number of components produce asymptotically efficient estimates for all a ∈ A simultaneously.

Design and analysis of splitting algorithms for rare event estimation

Paul Dupuis, Brown University, USA, [email protected]

Particle splitting methods are considered for the estimation of rare events with a large deviation scaling. A notion of subsolution is defined for a related calculus of variations problem, and two main results are de- scribed that hold under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution.

Session MA04 - Asymptotic Analysis 1, Chair: Bassamboo / Rand- hawa in CZ 11

Heavy Traffic Analysis of Maximum Pressure Policies for Stochastic Processing Net- works with Multiple Bottlenecks

Baris Ata, Northwestern University, Evanston, IL, USA, [email protected] Wuqin Lin, Northwestern University, Evanston, IL, USA, [email protected]

A class of open processing networks operating under a maximum pressure policy is considered in the heavy traffic regime. We prove that the diffusion-scaled workload process for a network with several bottleneck resources converges to a semimartingale reflecting Brownian motion (SRBM) living in a polyhedral cone. We also establish a state space collapse result that the queue length process can be lifted from the lower- dimensional workload process.

Slow-server problem in many-server heavy traffic with non-exponential service times

Tolga Tezcan, Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, [email protected]

We consider the slow-server problem with non-exponential service times. We assume that service times have a certain phase-type distribution. Specifically, service time of a server is assumed to have an ex- ponential distribution with rate µi with probability pi . Specific values of the parameters for each server depend on the server pool it belongs to. We show that a static priority policy is asymptotically optimal in the many-server heavy-traffic regime.

53 11 – Abstracts

Validity of heavy-traffic steady-state approximations in open queueing networks: sufficient conditions involving state-space collapse

I. Gurvich, Columbia University, New York, [email protected] A. Zeevi, Columbia University, New York, [email protected]

A class of diffusion processes known as semi-martingale reflecting Brownian motions (SRBMs) is often used to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson (1998) and Williams (1998) laid out a general and structured approach for proving the validity of such heavy-traffic approximations, in which the SRBM model is obtained as a diffusion limit from a suitably normalized queue-length process. The Brownian system model provides, among other things, a simplified means for approximating the steady-state behavior of the underlying queueing network. However, for the multiclass network models studied by Bramson and Williams, it is not clear whether there is rigorous justification for such an approx- imation. In this paper we give a broad set of sufficient conditions, building on the work of Bramson and Williams, under which the above question can be answered in the affirmative. The analysis illuminates some inter- esting connections between: stability of the pre- and post-limit processes; the fluid models corresponding to the two systems; state-space collapse; and the respective roles played by all of the above in establishing validity of heavy-traffic steady-state approximations.

Rental duration distributions and stock-levels: Why deadlines may be bad for rental firms

A. Bassamboo, Northwestern University, USA, [email protected] S. Kumar, Stanford University, USA, [email protected] R.S. Randhawa, The University of Texas at Austin, USA, [email protected]

We study a rental system where a fixed number of heterogeneous users rent one product at a time from a collection of re-usable products. The online DVD rental firm Netflix provides the motivation. We assume that rental durations of each user are i.i.d, non-lattice and have finite mean. We study transient behavior in this system following the introduction of a new product that is desired by all the users. We represent the usage process for this new product in terms of an empirical distribution. This allows us to characterize the asymptotic behavior of the usage process as the number of users increases without bound, via appropriate versions of Glivenko-Cantelli and DonskerÂ’s theorems. Analyzing the usage process, we demonstrate that an increase in the variability of the rental duration distribution can actually help the firm by allowing it to set lower capacity levels to provide a desired quality of service. Further, we show that the firm is better off not imposing any deadlines for the return of the product.

Session MA05 - Scheduling, Chair: Borst / Wierman in CZ 12

New Approximations and Bounds for an M/G/s FIFO queueing system

Varun Gupta, Carnegie Mellon University, [email protected] Jim Dai, Georgia Institute of Technology, [email protected] Mor Harchol-Balter, Carnegie Mellon University, [email protected] Bert Zwart, Georgia Institute of Technology, [email protected]

The M/G/s queueing system is the oldest multiserver system model, and has been the topic of performance papers for almost half a century. However, even now, only coarse approximations exist for its mean delay.

54 MA05

All the closed-form (non-numerical) approximations in the literature are based on the first two moments of the job size distribution. In this work we develop new techniques to obtain bounds and approximations for the mean delay of an M/G/s queueing system.

Optimal scheduling discipline in an M/G/1 queue with Pareto-type service time distributions

S. Aalto, Helsinki University of Technology, Finland, samuli.aalto@tkk.fi U. Ayesta, LAAS-CNRS, France, [email protected]

We consider the mean delay optimization in an M/G/1 queue for some Pareto type service time distri- butions. If the Pareto distribution is defined in such a way that all positive values are allowed, then the distribution has a decreasing hazard rate, which implies that the FB discipline is optimal. However, if it is defined starting from a strictly positive value, the hazard rate is no longer monotone. We show that, not only for this but for a whole class of service time distributions, the optimal discipline is a combination of FCFS and Foreground-Background (FB) disciplines, which gives full priority to the jobs with attained service less than some fixed threshold θ ∗. These priority jobs are served in the FCFS manner. If there are no jobs with attained service less than θ ∗, the full priority is given to the job with least amount of attained service. In our proof, we utilize the index approach developed by Gittins and Klimov. In addition to the ordinary Pareto type distributions with an infinite support, we also discuss the structure of the optimal scheduling discipline when dealing with bounded Pareto distribution with a finite support.

Revisiting the performance of large jobs in the M/GI/1

Adam Wierman, Carnegie Mellon University, USA, [email protected]

In recent years, the response times (sojourn times) experienced by large job sizes have been the focus of a large number of papers. Though results about many scheduling disciplines have appeared, to this point, results characterizing the response time of large job sizes have been limited to either mean value analysis or law of large numbers scalings. We will present a new framework that unifies these results and provides new results characterizing the distributional behavior of large job sizes. Also, we will illustrate the impact these new results have (i) for the analysis of the response time tail, (ii) for the analysis of busy periods, and (iii) for predicting response times of customers upon arrival.

Scheduling in polling systems

E.M.M. Winands, Eindhoven University of Technology, The Netherlands, [email protected] A.C. Wierman, Carnegie Mellon University, USA, [email protected] O.J. Boxma, Eindhoven University of Technology, The Netherlands, [email protected]

We present a simple mean value analysis (MVA) framework for analyzing the effect of scheduling within queues in classical asymmetric polling systems with gated or exhaustive service. Scheduling in polling systems finds many applications in computer and communication systems. Our framework leads not only to unification but also to extension of the literature studying scheduling in polling systems. It illustrates that a large class of scheduling policies behaves similarly in the exhaustive polling model and the standard M/GI/1 model, whereas scheduling policies in the gated polling model behave very differently than in an M/GI/1.

55 11 – Abstracts

Session MA06 - Call Centers 1, Chair: Koole in CZ 13

Density Forecasting of Intraday Call Center Arrivals using Models Based on Expo- nential Smoothing

J.W. Taylor, Said Business School, University of Oxford, UK, [email protected] R.D. Snyder, Monash University, Melbourne, Australia, [email protected]

Time series of intraday call center arrivals tend to consist of both intraweek and intraday seasonal cycles. In our previous work, we developed exponential smoothing formulations to accommodate the two seasonal cycles, with the emphasis being on point forecasting. In this paper, we adapt and extend these models for the univariate density forecasting of count data. The result is a number of alternative single source of error state space models with Poisson arrivals. We also consider single source of error state space quantile models, which have the appeal of avoiding distributional assumptions. Our empirical analysis evaluates point forecasting and density prediction up to two days ahead using call center arrivals for NHS Direct, the 24 hour telephone helpline provided by the National Health Service in England and Wales.

Short term dynamic prediction in Call Centers

G. Jongbloed, Delft University of Technology, The Netherlands, [email protected] J.L. Eveleens, VU University, The Netherlands, [email protected]

In call centers it is important to have reliable predictions for the number of incoming calls in future periods. For human resource management, a long term prediction of the work load is of interest. Within a shorter period, say a day, predictions are important when other tasks can be assigned to agents within the center. In this talk we consider prediction methods for the remaining number of incoming calls within the current day, given the observed numbers up till some time point during the current day. Also historical data on the process of incoming calls are available. First a stochastic model in the spirit of Whitt (1999) and Avramidis et al. (2003) is constructed and param- eters of interest are identified. An important parameter in the model is the distribution function G of the total number of calls for the present day. Avramidis et al. (2003) model this distribution using a Gamma mixture of Poisson variables in the spirit of Jongbloed and Koole (2001). We propose a more general and flexible model for the distribution of the number of daily incoming calls, a nonparametric mixture of Poisson distributions. This gain in flexibility is particularly useful in situations where there are unmeasured covariates that in fact define sub-populations of days.

References Avramidis, A.N., A. Deslauriers and L’Ecuyer. 2003. Modeling daily arrivals to a telephone call center. Management Science 50 896–908. Jongbloed, G. and G.M. Koole. 2001. Managing uncertainty in call centers using poisson mixtures. Ap- plied Stochastic Models in Business and Industry 17 307–318. Whitt, W. 1999. Dynamic staffing in a telephone call center aiming to immediately answer all calls. Oper- ation Research Letters 24 205–212.

Forecast errors in service systems

Samuel G. Steckley, The Mitre Corporation, U.S.A., [email protected] Shane G. Henderson, Cornell University, U.S.A., [email protected] Vijay Mehrotra, San Francisco State University, U.S.A., [email protected]

56 MA07

We investigate the presence and impact of forecast errors in the arrival rate of customers to a service system. Analysis of a large data set shows that forecast errors can be large relative to the fluctuations naturally expected in a Poisson process. We show that ignoring forecast errors typically leads to over-estimates of performance, and that forecast errors of the magnitude seen in our data set can have a practically significant impact on predictions of long-run performance. We also define short-run performance as the random percentage of calls received in a particular period that are answered in a timely fashion. We prove a central limit theorem that yields a normal-mixture approximation for its distribution for Markovian queues, and sketch an argument that shows that a normal-mixture approximation should be valid in great generality. Our results provide motivation for studying staffing strategies that are more flexible than the fixed-level staffing rules traditionally studied in the literature.

Shift Scheduling in Multi-skill Call Centers

Ger Koole, VU University Amsterdam, The Netherlands, [email protected]

We describe simple methods for shift scheduling in call centers. Existing methods for single-skill call centers are discussed and a new method for shift scheduling in multi-skill call centers is introduced. The method is easy to implement and has short computation times. A number of possible extensions to the models is discussed. For example, we explain how to include other communication channels besides telephone channels. Joint work with Sandjai Bhulai and Auke Pot.

Session MA07 - MDP 1, Chair: Feinberg in CZ 14

The entangled scheduling and routing problem

C.F. Bispo, Instituto de Sistemas e Robótica, Instituto Superior Técnico, Lisbon, Portugal, [email protected]

The majority of the basic building blocks constructed in the context of queuing theory is such that the centralized optimal decision policy, as computed by means of formulating a Markov Decision Problem (MDP), is non-idling when costs incurred are proportional to queue length. The consequences of this fact have been significant and profound in the research conducted for decades in this area. A basic queuing network will be presented which poses some interesting challenges. It is a combination of two classic decision problems in networks: the scheduling problem and the routing problem. For a network with three servers processing two customer classes one of the servers has to make scheduling decisions which have a routing consequence outside of its control. Unlike many basic toy problems, the distinguishing feature of the problem is the fact that, for some states, the optimal policy of this central server involves keeping it idle in the presence of customers. The problem will be formulated as an MDP and its optimal policy will be presented, as it is numerically produced for some instances of the problem. The consequences and challenges posed by this new problem, as well as its relevance in general queuing networks, will be discussed.

The power of deterministic open-loop control policies

D.A. van der Laan, Vrije University Amsterdam, The Netherlands, [email protected]

We investigate the control of a call center handling two types of incoming calls. A fixed number of C parallel homogeneous agents are available to respond incoming calls of both types. This can be modelled as controlling a discrete event dynamic system. We assume that type 1 (type 2) calls arrive at the system

57 11 – Abstracts

according to a Poisson process of rate λ1 (λ2) respectively. The time it takes for any agent to answer a type 1 (type 2) call is generally distributed with means 1 ( 1 ) respectively. We assume there is an infinite FIFO µ1 µ2 queue for waiting type 1 calls and there is a second infinite FIFO queue for waiting type 2 calls. Possible decision events occur both at moments of arrival and service completions. After a service completion the control policy should give a rule to decide whether the free agent should serve the (longest waiting) type 1 or type 2 call. The same applies for moments of arrival if there is at least one idle agent at that time. If at any decision event the free agent is assigned to a type of calls for which no one is waiting then the agent will be idling. Let E(W1) and E(W2) be the expected average waiting time for type 1 and type 2 calls respectively. Then the performance measure we consider will have the form A1 E(W1) + A2 E(W2) for some positive constants (weights) A1, A2. Typically reducing waiting times of type 1 jobs has much more priority than reducing waiting times of type 2 jobs and thus we put A1 = 10, A2 = 1 or something like that.

In case of exponentially distributed service times controlling the system may be modelled as a Markov Decision Problem (MDP) with a state space consisting of four integer components (x1, x2, x3, x4), where x1 (x2) is the number of waiting type 1 (type 2) jobs respectively and x3 (x4) is the number of agents serving type 1 (type 2) jobs. However, because of the curse of dimensionality solving this MDP by dynamic programming is awkward especially in case of a larger number of agents C. Moreover, this description is only possible if exponentially distributed service times are assumed while in real-life systems service times are more generally distributed. Finally the corresponding decision rules obtained by dynamic programming are state dependent and therefore hard to implement in a real-life call center. These are all arguments to look for a more robust (it should not be for only a particular case) approach and for that we consider control policies which are not state dependent. Such policies are called open-loop control policies and the well- known Bernoulli policies fall in this class. For systems as described above a Bernoulli policy assigns at any decision event the free agent to a type 1 call with probability θ and to a type 2 call with the remaining probability 1 − θ. Bernoulli policies are randomized policies and it is possible to optimize over θ by for example simulation and a binary search. However, instead of the Bernoulli (randomized) policies it is also possible to apply deterministic open-loop policies. The idea remains to optimize over θ, the long-run proportion of time an agent is assigned to a type 1 job, just as for Bernoulli policies. However, instead of throwing a "θ-coin" as Bernoulli policies do, a so-called balanced sequence of zeros and ones of rate θ is used which determines all assignments. Theoretical arguments for the superiority in performance of such a deterministic open-loop policy of rate θ compared to the corresponding Bernoulli policy of rate θ will be given. Moreover, we will confirm this by results from simulations in which we optimize over θ by looking at the corresponding simulated performances for both deterministic and Bernoulli control policies.

A MDP approach to the control of traffic lights

René Haijema, University of Amsterdam, The Netherlands, [email protected] Jan van der Wal, Eindhoven University of Technology, The Netherlands.

Optimal control of traffic lights, such that the overall average waiting time per car is minimized, appears to be quite complicated. For large (networks of) intersections, the MDP becomes intractable due to the curse of dimensionality in the state space. We suggest a one-step policy improvement approach based on the relative values of a (near-) optimal fixed cycle. Under the fixed cycle the value of a state at an intersection is decomposed into values for each queue. Key idea of our so-called relative value based rules is to interrupt the fixed cycle, by shortening or lengthening green periods and when all signals show red to prescribe which queue to serve next. We consider several relative value based rules and compare them to the underlying fixed cycle, and actuated control policies and for tractable cases also to the optimal (MDP) policy. The rules can be extended to deal with arrival information, and extensions to networks of intersections and appear to perform very well with waiting time reductions in the order of 20% or even more.

58 MA08

Splitting of randomized stationary policies in absorbing Markov Decision Processes

E.V. Denardo, Yale University, USA, [email protected] E.A. Feinberg, Stony Brook University, USA, [email protected] U.G. Rothblum, Technion, Israel, [email protected]

To split a randomized stationary policy is to construct a mixture of nonrandomized stationary policies with the same occupation measure as the original policy. We discuss absorbing Markov Decision Processes (MDPs) which are generalizations of discounted MDPs. For absorbing MDPs with Borel state and action sets, we present two results on splitting of a randomized stationary policy into a mixture of nonrandomized stationary policies. The first result is an explicit formula for the splitting at one state. The second result is a splitting of a K -randomized stationary policy into a mixture of (K + 1) stationary policies {φ1, i = 0, 1,..., K } such that the policies φi and φi+1 differ at most at one state, i = 0, 1,..., K − 1. These results are useful for studying of MDPs with multiple objectives and constraints. We discuss applications of these results to countable discounted MDPs and to constrained bandit problems.

Session MA08 - Retrial Queues, Chair: Artalejo in CZ 15

Cellular mobile radio telephone networks with priority active customers

F. Machihara, Tokyo Denki University, Japan, [email protected]

In cellular mobile radio telephone networks, hand-off failures produce retrials. In particular, networks be- ing congested, the influence upon the network performance becomes serious. We may give some priorities to mobile active customers over mobile inactive cutomers in order to protect against the deterioration. The first method is a circuit reservation system. For mobile active inflow customers into the tagged cell, a few circuits are reserved. The second method is a soft-hand-off system. Even if a mobile active customer who is getting into the tagged cell meets all circuits being busy, his service is never interrupted. He can continue to hold his circuit in the pre-existing cell, until a circuit becomes free in the tagged cell. We quantify the effect of these two methods.

A discrete-time queueing model with abandonments

Rein Nobel, Department of Econometrics, Vrije Universiteit, Amsterdam

This paper presents a one-server queueing model with abandonments in discrete time. In every time slot a generally distributed number (batch) of customers arrives. The different numbers of arrivals in consecutive slots are mutually independent. Each customer requires a generally distributed service time, counted in slots. Customers arriving in a slot can start their service only at the beginning of the next slot [i.e. delayed access]. When upon arrival customers find the server busy, they join a queue and wait for their service. When upon arrival customers find the server idle, then one of the incoming customers (randomly chosen) starts its service at the beginning of the next slot, whereas the other incoming customers, if any, join the queue. The customers in the queue are served in the order of arrival. Every time slot each customer waiting in the queue decides to abandon the system for ever with a fixed (abandon) probability, i.e. independently from the other customers waiting in the queue, his position in the queue or the elapsed waiting time. Arrivals have precedence over departures [i.e. late arrivals]. The steady-state behaviour of this system is studied.

M/M/s Retrial Queue with Balking and Reneging

Yang Woo Shin, Department of Statistics, Changwon National University, Changwon, Gyeongnam 641-773, KOREA, [email protected]

59 11 – Abstracts

Taek Sik Choo, Department of Statistics, Changwon National University, Changwon, Gyeongnam 641-773, KOREA

We consider an M/M/s retrial queue with balking and reneging as a call center model with impatient customers. The customer who has balked at entering the system or reneged on receiving service in service line may join the virtual pool of customers, called orbit and repeats its request after random amount of time. The behaviors of retrial customers from orbit are the similar to those of primary customers that arrive from outside. The system is described by a level-dependent-quasi-birth-and-death (LDQBD) process whose block matrix components are of infinite size. We derive an algorithmic solution for the number of customers in the system. The algorithm is based on the generalized truncation method in principle. However, due to the infinite size of block components, the truncation procedure is applied in two steps and the algorithm consists of computing the stationary distributions of two-node network and retrial queue with finite capacity in service facility.

Algorithmic analysis of the Geo/Geo/c retrial queue

J.R. Artalejo, Complutense University of Madrid, Spain, [email protected]

In this paper, we consider a discrete-time queue of Geo/Geo/c type with geometric repeated attempts. It is known that its continuous counterpart, namely the M/M/c queue with exponential retrials, is analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused from the retrial feature. In discrete-time, the ocurrence of multiple events at each slot increases the complexity of the model and raises further computational difficulties. We propose several algorithmic procedures for the effcient com- putation of the main performance measures of the system. More specifically, we investigate the stationary distribution of the system state, the busy period and the waiting time.

Session MA09 - Inventory 1, Chair: Van Houtum in CZ 16

Coordinated replenishment in a two echelon inventory system with transportation capacity

N.C. Buyukkaramikli, Eindhoven University of Technology, The Netherlands, [email protected] U. Gurler, Bilkent University, Turkey, [email protected] O. Alp, Bilkent University, Turkey, [email protected]

In this study, we examine a coordinated replenishment problem in the presence of transportation capacity. The aim is to minimize the expected cost rate function with respect to several system parameters. We first study a single-echelon setting with multiple retailers under a quantity based joint replenishment policy. A limited fleet of capacitated trucks is assumed for the transportation of the orders from the ample supplier, which is modeled as a queueing system. Operating characteristics are derived and the objective function is studied using numerical methods. The model is then extended to a two-echelon inventory system, where the warehouse employs an (S − 1, S) inventory policy. The departure process of the warehouse inventory system is characterized, which constitutes the arrival process of the transportation unit that delivers the warehouse orders to the retailers. This process is then approximated by an Erlang Process. Several nu- merical studies are conducted to assess the sensitivity of the expected total cost rate to the system and cost parameters as well as the performance of the approximation.

60 MA09

A multi-item inventory system with multiple resource constraints

Minmin Qiu,IBM China Research Laboratory, P.R.China, [email protected] Hongwei Ding,IBM China Research Laboratory, P.R.China, [email protected] Jin Dong,IBM China Research Laboratory, P.R.China, [email protected] Changrui Ren,IBM China Research Laboratory, P.R.China, [email protected] Wei Wang,IBM China Research Laboratory, P.R.China, [email protected]

Real life inventory systems are always capacitated by multiple resources such as storage areas and capitals invested. We model such an inventory system with multiple limited resources under (r,Q) inventory policy (inventory is reviewed continuously) or (s,S) policy (inventory is reviewed periodically). The demand for each item from customers is stochastic and stationary. All items in the inventory system are at unit form. Outside suppliers with infinite capacity replenish inventories to the system with constant lead time for each item. The optimization problem is to find the best inventory policies for each item, satisfying all resource constraints. We first solve a relaxed optimization problem to obtain the initial inventory policies for all items, regardless of resource constraints. For each solution, eight neighbor solutions are defined. Then a heuristic algorithm is applied to the original optimization problem to improve the initial solution by searching for the best neighbor solutions continuously. Numerical examples show that in most cases the algorithm generates high quality solutions. A case study with actual data from a large distribution center shows the cost can be reduced by 8.1% compared with current inventory policies.

Coordination in divergent multi-echelon inventory systems

J. Timmer, University of Twente, The Netherlands, [email protected]

This presentation is concerned with the coordination of inventory control in divergent multi-echelon inven- tory systems under decentralized control. All installations in these supply chains track echelon inventories. Under decentralized control the installations will decide upon base stock levels that minimize their own inventory costs. In general these levels do not coincide with the optimal base stock levels in the global optimum of the chain under centralized control. Hence, the total cost under decentralized control is larger than under centralized control. To remove this cost inefficiency, a coordination mechanism is presented. This mechanism results in the global optimum of the chain being the unique Nash equilibrium of the corre- sponding strategic game. Furthermore, all installations agree upon the use of this mechanism because they result in lower costs per installation.

A product-form solution for two-echelon spare parts networks with emergency ship- ments

G.J. van Houtum, Technische Universiteit Eindhoven, The Netherlands, [email protected] R.J. Boucherie, University of Twente, The Netherlands, [email protected] S.A. Heideveld, University of Twente, The Netherlands, [email protected]

We consider an inventory model for repairable parts for capital goods with high down time costs. We have one central warehouse, multiple local warehouses, and one central repair facility. When a part fails at a customer, a ready-for-use part is immediately sent from the closest local warehouse if possible. Simulta- neously, the failed part is sent into repair. If the local warehouse is out of stock, then, via an emergency shipment, a ready-for-use part is sent from the central warehouse if a part is available there, and otherwise from the central repair facility. We assume Poisson demand processes, generally distributed leadtimes for replenishments, repairs, and emergency shipments, and a basestock policy for the inventory control. Via a transformation into a network of Erlang loss queues with hierarchical jump-over blocking, we show that the steady-state behavior of our inventory system has a product-form solution. This main result has several important implications.

61 11 – Abstracts

Monday 2:00pm - 3:30pm

Session MB01 - Financial Engineering 2, Chair: Schoutens / Valdivieso in CZ 4

Optimal Portfolio Choice with Annuitization

Ralph S.J. Koijen,Finance Group, CentER, Tilburg University and Netspar, Tilburg, the Netherlands, [email protected] Theo E. Nijman, Finance and Econometrics Group, CentER, Tilburg University and Netspar, Tilburg, the Netherlands, [email protected] Bas J.M. Werker, Finance and Econometrics Group, CentER, Tilburg University and Netspar, Tilburg, the Netherlands, [email protected]

We study the optimal life-cycle consumption and portfolio choice problem taking into account annuity risk at retirement. Optimally, the investor allocates retirement wealth to nominal, inflation-linked, and variable annuities and conditions this choice on the state of the economy. However, we also consider often-used sub-optimal annuitization strategies. We determine subsequently how the investor optimally anticipates annuitization before retirement. We find that i) information on term structure variables and risk premia sig- nificantly improves the optimal annuity choice, ii) annuity market incompleteness is economically costly, and iii) adjustments in the optimal investment strategy before retirement induced by the annuity demand are significant.

Measuring comonotonicity in random vectors

I. Koch, University of Antwerp, Belgium, [email protected]

In the last decades we observe an increasing interest from researchers and practitioners in the dependence structure of a random vector. Many research is already performed in this field e.g. through the theory of comonotonicity. In fact the comonotonic dependence structure in particular has proved to be a very useful tool when approximating an unknown but strongly positively dependence structure. As a consequence of this evolution, there is a need for a measure reflecting how close a given dependence structure approaches the comonotonic one. In this talk we design the comonotonicty coefficient. This coefficient can be calculated for multivariate vectors and examines to what extent the considered vector is comonotonic. This measure takes values in the range [0, 1]. As we want to quantify the degree of comonotonicity, this measure is defined in such a way that it equals 1 in case the random vector is comonotonic and equals 0 in case it is independent. We show how such a measure can be designed analytically, by making use of copulas for the modeling of the dependence structure and we discuss the main properties of this comonotonicity coefficient.

A tractable multi-firm default model with jumps

Matthias Scherer, HVB-Institute for Mathematical Finance, TU Munich, [email protected]

We present a multi-firm structural-default model which allows the pricing of bonds, CDS, and CDOs in a consistent framework. Each firm-value process is modeled as the exponential of a jump-diffusion process, as proposed by Zhou (2001). Default is triggered by the first passage of the firm-value process below some default threshold. Dependence among the companies is introduced by means of a common market factor and by supporting common jumps. Jumps in the firm-value process of some company are triggered by a thinned-out Poisson processes. This construction preserves the distribution of each individual firm- value process, which is a crucial property for a calibration of the model. On a firm-individual basis, one

62 MB02 feature of our model are positive credit spreads for bonds and CDS with short maturities. This corrects a major shortfall of pure-diffusion models. Moreover, it turns out that the term-structure of credit spreads, as implied by the model, is extremely flexible and allows an excellent fit to observed prices. On a portfolio basis, allowing simultaneous jumps provides an additional parameter to adjust the dependence among the companies. However, the distribution of first-passage times is not known for arbitrary jump-diffusion processes. To evaluate bond and CDS prices in our model, we present an efficient Monte Carlo simulation and a numerical algorithm based on the Laplace transform of first-passage times. Such transforms are available under the assumption of two-sided exponentially distributed jumps, due to Kou and Wang (2003). Model prices of CDOs and nth-to default contracts can then be found by Monte Carlo simulations. Finally, we present a calibration of the model in two steps. At first, each individual firm-value process is calibrated to CDS quotes. Then, the parameters of the dependence structure are calibrated using CDO quotes.

Parameter estimation in fractionally integrated process of Ornstein-Uhlenbeck type

Luis Valdivieso, Katholieke Universiteit Leuven, Belgium, [email protected]

An estimation methodology to deal with fractionally integrated processes of Ornstein-Uhlenbeck type is proposed. The methodology is based on the continuous Whittle contrast. Simulations results are presented for a symmetric CGMY background driving Lévy process. An application to stochastic volatility modelling is also discussed.

Session MB02 - Percolation and Related Topics, Chair: Van den Berg in CZ 5

Exceptional times for the dynamical discrete web, part I + II

L.R.G. Fontes, Univ. of São Paulo, Brazil, [email protected] C.M. Newman, New York Univ., U.S.A., [email protected] E. Schertzer, New York Univ., U.S.A., [email protected] K. Ravishankar, SUNY New Paltz, U.S.A., [email protected]

Howitt and Warren introduced a system of coalescing one-dimensional simple symmetric random walks that evolve in an extra continuous dynamical parameter s. The evolution is by independent updating of the Bernoulli variables that describe the discrete web at any fixed s. Motivated by results of Häggstrom-Peres- Steif and Schramm-Steif on dynamical percolation, we have studied (ArXiv 0704.2706) exceptional times for this dynamical discrete web. One result is that there are√ exceptional values of s such that the walk from the origin Ss(n) is asymptotically bounded above by K n with a nontrivial dependence of the Hausdorff dimension of the set of such s on the parameter K . If time permits, I will discuss the dynamical Brownian web, a natural scaling limit of the dynamical discrete web.

Sharp phase transitions in divide-and-colour models

R.J. Meester Andras Balint Federico Camia

We study a dependent percolation model, in which subcritical percolation clusters (say with parameter q) are independently coloured with two colours, black and white. We show that the phase transition is sharp. 1 This implies that on the triangular lattice, the critical value for the colouring process is 2 , independently of q, as long as q is subcritical.

63 11 – Abstracts

Multiscale analysis of stochastic reaction networks

Thomas G. Kurtz, University of Wisconsin-Madison, Madison, WI, USA, [email protected]

Attempts to model chemical reactions within biological cells have led to renewed interest in stochastic models for these systems. These models of cellular chemical reaction networks typically involve chemical species numbers and reaction rates varying over several orders of magnitude. A number of researchers have proposed exploiting the“multiscale” nature of these models to reduce the complexity of the model to be analyzed or simulated. Using elementary understanding of some basic stochastic processes, systematic approaches to model reduction will be discussed.

Session MB03 - Simulation 1, Chair: Henderson in CZ 10

Bandits and simulation selection options

S.E. Chick, INSEAD, France, [email protected] N. Gans, The Wharton School, The United States of America, [email protected]

This talk examines new work in the area of selecting the best of a finite set of alternatives, where best is the maximum mean performance, and the mean is unknown but inferable from stochastic simulation experiments. Unlike the usual approaches, which provide frequentist or Bayesian statistical properties, or which provide probabilistic convergence results, we link the economic costs (of sampling and time discounting) of simulation experiments to the expected NPV of system implementation decisions that are implied by the simulation analysis. For some special cases, we show how to express this problem as a stoppable bandit process, prove that a Gittins index policy is optimal, and develop diffusion approximations for the optimal allocation indices.

On transience and recurrence in discrete-event simulations

P.J. Haas, IBM Almaden Research Center, USA, [email protected] P.W. Glynn, Stanford University, USA, [email protected]

The generalized semi-Markov process (GSMP) is the usual model for the underlying stochastic process of a complex discrete-event simulation. It is important to understand fundamental behavioral properties of the GSMP model, such as the conditions under which the states of a GSMP are recurrent. For example, the recurrence or non-recurrence of a specified “single state” determines whether or not the successive exit times from the state can be used as a sequence of regeneration points. We review some sufficient conditions for recurrence in irreducible finite-state GSMPs; these conditions include requirements on the "clocks" that govern the occurrence of state transitions. For example, each clock-setting distribution must have finite mean. We then show that, in contrast to ordinary semi-Markov processes, an irreducible finite-state GSMP can have transient states in the presence of clock-setting distributions with heavy tails.

Analysis Of State-Dependent Importance Sampling Algorithms

Jose Blanchet, [email protected]

Importance Sampling (IS) is a popular variance reduction technique in simulation applications and is par- ticularly useful in rare-event simulation. Nevertheless, as several results and examples developed recently indicate, (IS) must be applied with care in the performance analysis via simulation of stochastic systems. In particular, it is often the case that one must resort to State-dependent Importance Sampling (SDIS) to obtain reliable estimators. In this talk, we discuss new techniques that allow both to design efficient

64 MB04

SDIS estimators and theoretically verify their efficiency. We illustrate the techniques, which are based on Lyapunov-type inequalities, with examples drawn from combinatorics (counting graphs), queueing theory and risk analysis.

Adaptive control variates for steady-state simulation

S. Kim, Purdue University, U.S.A., [email protected] Shane G. Henderson, Cornell University, U.S.A., [email protected]

We introduce an adaptive control variate scheme for estimating steady-state performance measures for stochastic systems using simulation. We use a procedure based on sample average approximation for tuning the control variate estimators. The key assumption in our analysis is that the underlying process is regenerative.

Session MB04 - Stability of Stochastic Networks, Chair: Ramanan in CZ 11

Steady-state analysis of a multi-server queueing system in QED regime

D. Gamarnik, MIT, USA, [email protected] P. Momcilovic, University of Michigan Ann Arbor, USA, [email protected]

We consider a multi-server GI/GI/N queueing system in Halfin-Whitt (QED) regime. The analysis of this system for the case of non-exponentially distributed service times presents a significant challenge. We consider the case of discrete service time distributions and show that in the large N limit the queue length process can be approximated by a simple discrete time continuous state Markov chain. We then establish an interchange of steady state and large N limits using tightness arguments. We further derive a simple expression for the decay rate of the limiting queue length process.

The heavy traffic limit of an unbalanced generalized processor sharing model

Kavita Ramanan, Carnegie Mellon University, Pittsburgh, PA, [email protected] Martin I. Reiman, Bell Labs, Alcatel-Lucent, Murray Hill, NJ, [email protected]

We consider a server that processes J classes using the generalized processor sharing discipline. Extending a previous paper where we assumed that the work arrival rate and nominal service rate for each class are equal, we consider the situation where the total arrival rate and service rate are equal, but the rates for individual classes may not be. This gives rise to state space collapse. The dimension of the limit process is obtained by determining the invariant manifold of the fluid limit associated with this model. We show that when the heavy traffic condition holds, the functional central limit of the scaled unfinished work process is a reflected diffusion process that lies in the invariant manifold. The reflected diffusion limit is characterized by the extended Skorokhod map and may fail to be a semimartingale.

Fluid model for a data network with alpha fair bandwidth sharing and general doc- ument size distributions: two examples of stability

H.C. Gromoll, University of Virginia, USA, [email protected] N.H. Lee, University of California at San Diego, USA, [email protected] R.J. Williams, University of California at San Diego, USA, [email protected]

65 11 – Abstracts

We consider a fluid model for a data network with alpha fair bandwidth sharing and general document size distributions. This fluid model uses a measure valued state descriptor. The invariant states for the fluid model are characterized. Two examples are given of network topologies for which stability of the fluid model can be established under a nominal condition. The two types of networks are linear networks and tree networks. The latter is particularly interesting since for the original data network model, the distribution of the number of documents process in steady state is expected to be sensitive to the (non- exponential) document size distribution. Related work on using the fluid model to study stability of the original data network is also described.

Stability of processor sharing networks with simultaneous resource requirements

J. Hansen, Heriot-Watt University, Edinburgh, UK, [email protected] C. Reynolds, Heriot-Watt University, Edinburgh, UK, [email protected] S. Zachary, Heriot-Watt University, Edinburgh, UK, [email protected]

We study the phenomenon of entrainment in processor sharing networks, whereby, while individual net- work resources have sufficient capacity to meet demand, the requirement for simultaneous availability of resources means that a network may nevertheless be unstable. We show that instability occurs through poor control, and that, for a variety of network topologies, only small modifications to controls are required in order to ensure stability. For controls which possess a natural monotonicity property, we give some new results for the classification of the corresponding Markov processes, which lead to conditions both for stability and for instability.

Session MB05 - Communication Systems 1, Chair: Choi in CZ 12

Performance Analysis of IEEE 802.11e EDCA in Non-saturation Condition by M/G/1 Queue

Y.H. Bae, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] K.J. Kim, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] B.D. Choi, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected]

We study performance analysis of IEEE 802.11e EDCA in non-saturation condition where 4 traffic classes are served. There have not been many analytic works in non-saturation mode due to analytic complexity of models. However, non-saturation mode is valid in most real WLAN networks. We deal with the non- saturation case that packets arrive according to Poisson process and queue up to a buffer at a station. We derive probability generating function of HoL-delay Si of traffic class i. We model a station with traffic class i as M/G/1 queueing system where the service time is the HoL-delay Si , and then obtain the several performance measures such as total access delay and throughput by applying the theory of M/G/1 queue.

Performance Analysis of IEEE 802.15.4 Unslotted CSMA/CA

Tae Ok Kim, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] Jin Soo Park, USN Service Division, KT, Seoul,Korea, [email protected] Kyung Jae Kim, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected]

66 MB05

Bong Dae Choi, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected]

The standard of IEEE 802.15.4 MAC layer adopts beacon mode and non-beacon mode. Non-beacon mode is operated by unslotted CSMA/CA algorithm. This paper is the first work on performance of IEEE 802.15.4 non-beacon mode. We develop a new method using the theory of regenerative process. We obtain performance measures such as throughput, packet delay, packet loss probability and energy con- sumption. We show that numerical result is quite close to simulation result. Our performance measures are used for determining the optimal number of devices while supporting the required QoS constraints and for calculating the life time of battery.

Performance analysis of IEEE 802.15.4 Non-beacon Mode by Piggyback Method

Jin Soo Park, USN Service Division, KT, Seoul,Korea, [email protected] Tae Ok Kim, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] Kyung Jae Kim, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] Bong Dae Choi, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected]

We analyze the MAC performance of the IEEE 802.15.4 LR-WPAN non-beacon mode with piggyback download method in non-saturated condition. Our approach is to model stochastic behavior of one device as a discrete time Markov chain. First, we propose an analytical model describing the download behavior of a device using piggyback method. Then, by combining the model of a device with upload traffic in [3] and one with download traffic in this paper, we obtain the performance measures such as throughput, packet delay, energy consumption and packet loss probability of a device with both upload and download traffic. Numerical results and simulation results show that by removing a backoff procedure in the backoff method, the piggyback method can reduce the delay, loss probability and energy consumption compared with backoff method. Our results can be used to find optimal number of devices with some constraints on these measures.

Performance Analysis of Session-Level Load Balancing Algorithms

Dennis Roubos, Vrije Universiteit Amsterdam, Faculty of Sciences, [email protected] Sandjai Bhulai, Vrije Universiteit Amsterdam, Faculty of Sciences, [email protected] Rob van der Mei, Vrije Universiteit Amsterdam, Faculty of Sciences, [email protected]

Load balancing (LB) is crucial for the efficient operation of big server clusters. In the past, many different LB strategies on the request level have been developed with great effectiveness in parallel applications. However, the LB problem is not yet solved completely; new applications and architectures require new features. In particular, secure environments require that LB is done at session level instead of request level; that is, once a session has been assigned to a server, all subsequent service requests are directed to the assigned server. Despite the fact that many commercial products have been brought to the market to implement LB at the session level, little insight has been obtained into the efficiency of such session-level LB algorithms, leaving ample room for performance improvement and optimization. Motivated by this, we study session-level LB with a focus on algorithms that are simple and easy to im- plement in real systems. The performance of the load balancer is highly dependent on the request profiles of the different sessions and the information that is available for decision making. We make this trade-off between the information that is available to the load balancer and the efficiency of the algorithm explicit by developing new algorithms, and compare their efficiency with existing algorithms. The algorithms are mainly based on the load of each server and the number of active sessions running on them. Extensive

67 11 – Abstracts validation in an experimental setting shows that our algorithms outperform the existing ones, and as such, provide a simple, easy-to-implement yet effective means to improve the efficiency of large server clusters.

Session MB06 - Many-server Models 1, Chair: Armony in CZ 13

Optimal Control of Parallel Server Systems with Many Servers

Jim Dai, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0205, [email protected] Tolga Tezcan, Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, [email protected]

We consider parallel server systems that consist of several customer classes and server pools with expo- nential service times. We propose a control policy that dynamically routes customers. The policy uses minimal state information; in particular, it does not require arrival rate information. In the Hafin-Whitt many-server heavy traffic regime, we show that this policy is asymptotically optimal to minimize the total linear holding and reneging costs. A key to the optimality proof is a state space collapse (SSC) result. We show how a framework of Bramson (1998) can be adapted, from conventional heavy traffic to many-server heavy traffic, to prove the SSC result.

Fill rate constraint compliance and the many-machine repairman problem

O. Jennings, [email protected] Isilay Talay Degrimenci

The classic repairman problem with spares consists of a fixed number of failure-prone machines, clients who use them, and repairmen responsible for fixing the broken machines. We assume that when their current machine breaks, clients turn in the machine and desire an immediate replacement. Using a many- machine / many-repairman asymptotic approach, we consider how one would provision and staff such a system so as to meet a target machine-replacement fill rate constraint.

Fair Dynamic Routing Policies in Large-Scale Service Systems with Heterogeneous Servers

A.R. Ward, University of Southern California, [email protected] Mor Armony, Stern School of Business, NYU, 44 West 4th Street, NY, NY 10012, [email protected]

Modern call centers typically have many agents answering calls with different skill levels. One natural control issue that arises is how to route customers to agents. It has been shown that routing customers to the fastest agent first asymptotically minimizes steady-state customer waiting times. Unfortunately, this policy penalizes the faster agents by keeping them busy at all times, and only allowing the slower agents to idle. We address the question of how to optimize system performance subject to fairness with respect to server utilization.

Large cross selling in call centers

Itay Gurvich, Columbia Business School, 4I Uris Hall, 3022 Broadway, NY, NY 10027, [email protected]

68 MB07

Mor Armony, Stern School of Business, NYU, 44 West 4th Street, NY, NY 10012, [email protected] Constantinos Maglaras, Columbia Business School, 409 Uris Hall, 3022 Broadway, NY, NY 10027, [email protected]

Cross selling is an increasingly prevalent practice in call centers. This is largely because it allows firms to dynamically segment callers and customize product offerings accordingly. In this work we develop a framework under which both marketing decisions (such as: customer segmentation and pricing) and oper- ational decisions (such as: staffing, call sequencing and dynamic cross selling) may be considered jointly. We propose near-optimal pricing, staffing, sequencing and cross-selling policies under various forms of customer segmentation. Finally, we characterize how the degree and timing of customer segmentation impacts these policies and, correspondingly, the center’s profitability.

Session MB07 - Games 1, Chair: Economou in CZ 14

Equilibrium customer strategies under various levels of information for the single server queue with unreliable server.

A. Economou, University of Athens, Department of Mathematics, Section of Statistics and Operations Research, Greece, [email protected] S. Kanta, University of Athens, Department of Mathematics, Section of Statistics and Operations Research, Greece, [email protected]

We consider the single server Markovian queue with an unreliable server, that is a server that alternates between active and idle (repair) periods. The arriving customers decide whether to join or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine customers’ behavior under various levels of information regarding the state of the system at arrival instances. More specifically, a customer may know or not know the state of the server and/or the number of present customers upon his arrival. We derive equilibrium strategies for the customers under the various information levels and we study the effect of the information to the system performance.

On the use of quasi birth-and-death processes in equilibrium selection in game the- ory and queueing networks

Madjid Amir, Sultan Qaboos University, Oman, [email protected], P.O. Box 36, PC 123, DOMAS/SQU, Al Khod, OMAN

Quasi birth-and-death processes, a natural multi dimensional counterpart of the usual birth-death processes, are known to differ notably from the latter due to the fact that they fail to satisfy the so called detailed balance equations. These processes are NOT reversible. I will show how these processes have successfully been used to investigate the crucial issue of equilibrium selection in Games. Furthermore, I intend to show the use of quasi birth-and-death processes to investigate both the transient and long-run behavior of certain processes pertaining to Queueing systems.

An analysis of oligopolistic competitive markets for queued services with QoS guar- antees

Parijat Dube, IBM T. J. Watson Research Center, Hawthorne, NY, [email protected] Rahul Jain, IBM T. J. Watson Research Center, Hawthorne, NY, [email protected] Corinne Touati, INRIA - LIG, Grenoble, France, [email protected]

69 11 – Abstracts

We consider the problem where competing providers offer queued services with QoS guarantees. Users have quasi-linear utility functions that depends on delay and the price paid. We first analyze the equilibrium market shares of various providers. We assume that each provider has a service queue (such as an M/M/1 FIFO or an M/G/1 processor sharing) with a given capacity. Each provider offers delay guarantees and announces prices. Its objective is to maximize its revenue. We first observe that a competitive equilibrium does not exist in such a market in general. However, we show that the (price, delay) pair offered by a provider is unique. We then model the interaction between the providers as a Stackelberg game. Our analysis seems to indicate non-existence of an equilibrium of this game. In fact, we observe a “price- war" phenomenon also seen in many other contexts. However, we conjecture that the allocations that the providers offer in the Stackelberg game model, are in the core of the competitive exchange economy. This is part of on-going work.

Optimal balking strategies in the single server Markovian queue with compart- mented waiting space.

A. Economou, University of Athens, Department of Mathematics, Section of Statistics and Operations Research, Greece, [email protected] S. Kanta, University of Athens, Department of Mathematics, Section of Statistics and Operations Research, Greece, [email protected]

We consider the single server Markovian queue and we assume that arriving customers decide whether to join or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We suppose that the waiting space of the system is partitioned in compartments of fixed capacity for a customers. We suppose that customers have partial information about the system. More specifically, the information is either the number of the compartment in which they enter or just the position within it. We examine customers’ behavior under the various levels of information and we identify equilibrium threshold strategies, thus extending the fundamental results of Naor and Edelson, Hildebrand who studied the equilibrium strategies in the observable and unobservable M/M/1 queues. The social and profit optimization problems are also solved.

Session MB08 - Fluid Queues, Chair: Adan in CZ 15

On the stationary measure of a reflected Brownian motion in a wedge: some explicit results

A.B. Dieker, University College Cork, Ireland, [email protected] J. Moriarty, University College Cork, Ireland, [email protected]

Although the stationary distribution of one-dimensional reflected Brownian motion with drift is exponen- tial, this is generally not true in a multidimensional setting. M. Harrison and R. Williams have characterized the class of reflected Brownian motions with exponential densities in terms of a so-called skew symmetry condition on the reflection directions. However, not much is known on the stationary measure (or its density) when the skew symmetry condition fails to hold. For special multidimensional reflected Brownian motions, an intriguingly simple formula arises from connections with reflection groups, which is a topic in algebra. In this formula, the stationary density is represented as a finite sum of exponential terms. This talk reports an attempt to marry this formula with the literature on two-dimensional reflected Brownian motion in a wedge.

70 MB09

Transient Fluid Solutions and Queueing Networks with Infinite Virtual Queues

Y. Nazarathy, The University of Haifa, Israel, [email protected] G. Weiss, The University of Haifa, Israel, [email protected]

Fluid solutions of queueing networks over a finite time horizon often involve non-empty queues and full utilization of some of the resources for substantial portions of the time horizon. This observation makes it appropriate to treat the queueing network for which the transient fluid solution is specified, as a network with infinite virtual queues (i.e. queues that never empty yet need to be served at specified nominal rates). Examples of such queueing networks have been previously analyzed under steady state by Weiss, Kopzon and Adan. As an example of this interaction between the fluid solution and the queueing network, we present a method for controlling a multi-class queueing network over a finite time horizon such that the holding R T PK costs ( 0 k=1 ck Qk(t)dt ) are approximately minimized by tracking an optimal fluid solution. Our method is based on an adaptation of Dai and Lin’s results regarding rate stability of the maximum pressure policy for stochastic processing networks with ρ ≤ 1 to the case of a multi-class queueing network with infinite virtual queues. We present the proof for the asymptotic optimality of our fluid tracking policy.

R, G and U and the time-inhomogeneous periodic Quasi-Birth Death Process

B.H. Margolius, Cleveland State University, Cleveland, OH, USA, [email protected]

We solve for the asymptotic periodic distribution and moments of the continuous time quasi-birth-and- death process with time-varying periodic rates in terms of R, G and U matrix functions which are analogues of the R, G and U matrices of matrix analytic methods. We evaluate these QBDs numerically by matching the moments of the distribution of a generalized random walk with the same transition probabilities as the QBD, but with no boundary at level zero, and then use the approximate distribution for the generalized random walk to evaluate the R, G and U matrix functions.

Insurance Risk with Variable Number of Policies

Ivo Adan, Eindhoven University of Technology, The Netherlands, [email protected] Vidyadhar Kulkarni, University of North-Carolina, [email protected]

In this talk we consider an insurance company selling life insurance policies. New policies are sold at random points in time, and each policy stays active for a random amount of time, during which the poli- cyholder pays premiums continuously at rate r. When the policy expires, the insurance company pays a claim of random size. The aim is to compute the probability of eventual ruin starting with a given number of policies and a given level of insurance reserves. We establish a remarkable result that, if the lifetimes of policies are i.i.d. exponential random variables with rate µ, then the ruin probability is identical to the one in the standard compound Poisson model where the reserves increase at constant rate r and claims occur according to a Poisson process with rate µ. We conclude that the ruin probability does not depend on the initial number of active policies, nor on the arrival process of new policies.

Session MB09 - Search Models, Chair: Klafter in CZ 16

Intermittent search processes

Olivier Benichou, CNRS, Paris VI, [email protected]

71 11 – Abstracts

Many physical, chemical or biological problems can be rephrased as search processes, involving a searcher and a target of unknown position. Intermittent search strategies, alternating active search phases and non- reactive displacement phases, are observed in various situations, ranging from the microscopic scale (as for the search for a specific sequence on DNA) to the macroscopic scale (animals searching for food). Using representative analytical models, we show that these intermittent strategies are efficient and determine which durations of each phase optimize the search time.

Search research

Gleb Oshanin, CNRS, Paris VI, [email protected]

We overview several recent results on search kinetics of an immobile target by a concentration of randomly moving searchers. The searchers perform intermittent random walks in discrete time n = 0, 1, 2, ..., N, where N is the maximal time the search process is allowed to run. With probability α the searchers step on a nearest-neighbor, and with probability (1 − α) they leave the lattice and stay off until they land back on the lattice at a fixed distance L away from the departure point. We calculate the probability PN that the target remains undetected up to the maximal search time N, and seek to minimize this probability with respect to α and L. We find that PN is a non-monotonic function of α, and show that there is an optimal choice αopt(N) of α well within the intermittent regime, 0 < αopt(N) < 1, whereby PN can be orders of magnitude smaller compared to the "pure" random walk cases α = 0 and α = 1. Moreover, we show that if we are at liberty to choose L, we may further decrease PN by orders of magnitude by choosing an optimal value of this parameter.

Parallel search of long circular strands: Modeling, analysis, and optimization

Iddo Eliazar, Holon Institute of Technology and Tel Aviv University, Israel, [email protected]

We introduce and explore a model of an ensemble of enzymes searching, in parallel, a circular DNA strand for a target-site. The agents performing the search combine local scanning - conducted by a 1D motion along the strand, and random relocations on the strand - conducted via a confined motion in the medium containing the strand. Both the local scan mechanism and the relocation mechanism are considered general. The search-durations are analyzed, and their limiting probability distributions - for long strands - are obtained in closed-form. The results obtained: (i) encompass the cases of single, parallel and massively- parallel searches, taking place in the presence of either finite-mean or heavy-tailed relocation times; (ii) are applicable to a wide spectrum of local scan mechanisms including linear, Brownian, selfsimilar (e.g., fractional Brownian and fractional Levy), and sub-diffusive motions; (iii) provide a quantitative theoretical justification for the necessity of the relocation mechanism; and, (iv) facilitate the derivation of optimal relocation strategies.

Stochasticity and search in gene regulation processes

Ralf Metzler, Physics Dept, University of Ottawa, 150 Louis Pasteur, Ottawa, ON, K1N 6N5, Canada, [email protected]

The search of DNA-binding proteins for their specific binding site on the DNA molecule requires the scan- ning of megabases of the genetic code. Up until now, the detailed mechanisms are quite poorly understood. Recent developments in single molecule experimental techniques make it possible to study the search pro- cess of these proteins in real time. I discuss on the background of these experiments the stochastic processes involved in the search, namely, 3D bulk diffusion, 1D sliding motion along the DNA, and intersegmental transfers. The latter, under typical in vitro conditions, involve long-tailed jump lengths leading to a Levy flight. It is discussed how Levy statistics improve the search efficiency in that system. However, in vivo,

72 MC01 the situation becomes more complicated, as the bulk motion due to molecular crowding by other macro- molecules in the cell has been shown to be subdiffusive. A simple model is discussed to investigate the effects of subdiffusion to the target search, and it is argued that a weak ergodicity breaking arises that may be of advantage for the search process. References:

[1] M. Lomholt, T. Ambjoernsson, and R. Metzler, Phys. Rev. Lett. 95, 260603 (2005). [2] I. M. Sokolov, R. Metzler, K. Pant, and M. C. Williams, Biophys. J. 89, 895 (2005). [3] M. Lomholt, I. M. Zaid, and R. Metzler, Phys. Rev. Lett. 98, 200603 (2007).

Monday 4:00pm - 5:30pm

Session MC01 - Exit Times, Chair: Rolski in CZ 4

On an exit problem of a two-dimensional risk process from the positive quadrant

Florin Avram, Pau University, France, [email protected] Zbigniew Palmowski, University of Wrocław, Poland, [email protected] Martijn Pistorius, King’s College London, UK, [email protected]

Consider two insurance companies that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem consid- ered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process we obtain a closed form expression for the ultimate ruin probability. In the general case we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity.

Gaussian storage processes: long-time behavior

K. De¸bicki, University of Wrocław, Poland, [email protected]

Consider a fluid queue with service rate c > 0 and input process modelled by a centered Gaussian process with stationary increments {η(t) : t ∈ R}. Let {W(t) : t ≥ 0} := {sups≤t (η(t) − η(s) − c(t − s)) : t ≥ 0} be the stationary buffer content process (storage process) for the system. We focus on the asymptotic analysis of the tail distribution of P( sup W(t) > u) t∈[0,T (u)] as u → ∞. It appears that the presence of short range dependence or long range dependence for η(·) significantly influences the asymptotics.

Additionally, assuming that η(t) = BH (t) is a fractional Brownian motion with Hurst parameter H, we obtain an Erdös-Révész type of law of iterated logarithm for W(t).

De Finetti’s dividend problem for a general Lévy insurance risk process

Z. Palmowski, University of Wrocław, Poland, [email protected] Florin Avram, Universite de Pau, France, [email protected] A. Kyprianou, The University of Bath, UK, [email protected] Martijn Pistorius, King’s College London, UK, [email protected]

73 11 – Abstracts

In this talk we consider the solution to the classical control problem due to De Finetti (1957) and Gerber (1969) which concerns the optimal payment of dividends from an insurance risk process prior to ruin. The calculations we present go much further than existing literature in that our calculations are valid for a general spectrally negative Lévy risk process as opposed to the classical Cramér-Lundberg process. Related is the problem where we impose the restriction that ruin be prevented by bail-out loans. Drawing on the fluctuation theory of spectrally negative Lévy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal amongst all admissible ones takes the form of a barrier strategy. We also calculate distributional properties of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The technique we use to derive these quantities appeals principally to excursion theory rather than integro-differential equations and makes a new link with the distribution of integrated exponential subordinators. Finally, we also consider the problem with fixed cost for taking out dividends.

The exact asymptotics of the time to collision tail distribution

Z. Puchała, Wroclaw University, Poland, [email protected] T. Rolski, Wroclaw University, Poland, [email protected]

n 1 n Let W = {x ∈ IR : x1 < . . . < xn} and Xt ,..., Xt be independent, skip-free, strong Markov processes = 1 n ∈ such that x (X0,..., X0 ) W. By skip-free we mean that the exit of Xt from W must be through W. We study the collision time τ = inf{t > 0 : Xt ∈/ W}. In the talk we discuss the following issues:

1. Karlin-McGregor theory for the distribution of τ for processes with the same transition law,

1 n 2. The exact asymptotics for IPx(τ > t) for Xt ,..., Xt being independent Poisson processes, contin- uous time random walks or standard Brownian motions with identical transition law,

1 n 3. The exact asymptotics for IPx(τ > t) for Xt ,..., Xt being independent Brownian motions with different drifts.

Examples to Young tableaux and Markovian queues will be mentioned.

Session MC02 - Interacting Particle Systems, Chair: Van den Berg in CZ 5

The super-process limit of the critical spread-out contact process above 4 spatial dimensions

Remco van der Hofstad, Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, [email protected]

d We consider the contact process on Z × R+, at the critical infection rate λc, for d > 4. The model is a “spread-out” model having long range parameterized by L ≥ 1. We consider configurations in which the cluster of the origin survives to time t, and scale space by t1/2. We prove that for L sufficiently large all the moment measures converge, as t → ∞, to those of the canonical measure of super-Brownian motion. We use lace expansion methods on the discretized contact process, which is an oriented percolation model. The proofs are adaptations of those for oriented percolation. This is joint work with Akira Sakai.

74 MC03

Duality and exact correlations in a model of heat conduction

C. Giardina, Eindhoven University of Technology, The Netherlands J. Kurchan, CNRS-ESPCI, Paris F. Redig, Mathematical Institute, Leiden University, The Netherlands

We study a stochastic model of heat conduction, recently introduced in the study of Fourier law. We discuss duality and show uniqueness and detailed properties of the non-equilibrium steady state, such as an exact expression of the two-point correlation function. Some similarities with the simple symmetric exclusion process are discussed.

Approximate zero-one laws, spin-flip dynamics and Ising percolation

J. van den Berg, CWI and Vrije Universiteit, Amsterdam

In 1993 Higuchi showed that Ising percolation on the square lattice has a sharp transition. By using ap- proximate zero-one laws and the Ising spin flip dynamics, we show how this result fits as a special case in a more general framework. This more general framework involves models where the spin values can be suitably ‘finitarily’ represented by i.i.d. random variables.

Session MC03 - Stochastic Programming and Optimization 1, Chair: Gunawan in CZ 10

Performance Evaluation of Interconnection Network System Based on Minimum Cut Sets Method

Indra Gunawan, Department of Mechanical and Production Engineering Auckland University of Technology, New Zealand, [email protected]

Interconnection network system consists of layers of switching elements connected together in a prede- fined topology providing the connectivity between the set of processors and the set of memory modules. Performance evaluation of network system is important as it determines the usability and efficiency of the network to provide services. A shuffle-exchange network (SEN), a specific type of Multistage Intercon- nection Network (MIN) that consists of 2x2 switching elements, has been widely considered as a practical interconnection system. This paper presents network reliability model based on minimum cut sets for SEN systems. A general equation for large-scale network reliability is derived subsequently to reduce the complexity of the reliability analysis for the interconnection network system.

Scheduling assembly flow-shops with resource leveling: a GA based heuristic ap- proach

Qinhua Wang, 1 - IBM China Research Laboratory, 2 - Department of Industrial Engineering, Tsinghua University, P.R. China, [email protected] Changrui Ren, Jin Dong, Hongwei Ding, Wei Wang, Minmin Qiu, IBM China Research Laboratory, P.R. China, {rencr, dongjin, dinghw, wangwcrl, qiumm}@cn.ibm.com

For large scale assembly manufacturing, global optimization approaches are time consuming and impracti- cable. A Genetic Algorithm (GA) based heuristic is proposed to make schedule for large scale of assembly tasks and level the resource requirement through scheduling operations in each working area separately. The variance of daily resource requirement is considered as the index for resource leveling. Firstly, a rolling

75 11 – Abstracts back method is applied to ensure all orders are finished between their release dates and the due dates while selecting corresponding operations as a group of scheduling targets in each step. Then the precedence and float interval are calculated for each operation through improved critical path method considering the re- markable constraints of parallel orders and sub-assembly relationships in assembly manufacturing. At last each group of operations is scheduled via GA with feasibility validation and the labor resource is leveled in the corresponding makespan. Computational tests are carried out on three levels of actual data and demon- strate that the algorithm is much more efficient than global optimization, and for each production unit, the variance of daily resource requirement decreases while there is also increase in global resource leveling compared with experience-based and principle-based schedule result generated by simple heuristics.

A Fast Entropy Method for Solving Integer Programs: Part I

R.Y. Rubinstein, Technion, Israel, [email protected]

We present a new method based on the classic minimum cross-entropy (MinxEnt) method, for approxi- mating a broad class of NP-hard integer programs including combinatorial optimization problems (COP’s) . The main idea of our approach is to associate with each COP problem an auxiliary convex MinxEnt program, which is easy to solve analytically and which has the so-called decomposability property in the sense that its optimal solution, the joint probability density function (pdf) presents a product of the marginal pdf’s. We show that the optimal marginal pdf’s converge to their associated degenerated counterparts. This allows to approximate the solution of COP’s deterministically, that is without resorting to simulation. The complexity of our method is defined by the complexity of the MinxEnt program, that is by complexity of a convex program with logarithmic objective function and linear constraints, the so-called log-linear con- vex program. Because of that our method allows to solve combinatorial optimization with thousands of variables and thousands of constraints. Supportive numerical results are presented.

A Fast Entropy Method for Solving NP-Hard Counting Problems: Part II

R.Y. Rubinstein, Technion, Israel, [email protected]

We present a new method based on the classic minimum cross-entropy (MinxEnt) method for approximat- ing a broad class of NP-hard counting problems, like knapsack, satisfiability and cliques. The main idea of our approach is to associate with each counting problem a convex MinxEnt program, which is easy to solve analytically and which has the so-called decomposability property in the sense that its optimal solution, the joint probability density function (pdf) presents a product of the marginal pdf’s. To approximate the desired counting quantity we use two approaches: Sanov’s theorem and importance sampling. In the former case the desired counting quantity is approximated deterministically, that is without resorting to simulation. The complexity of our method is defined by the complexity of the MinxEnt program, that is by complexity of a convex program with logarithmic objective function and linear constraints, the so-called log-linear convex program. Because of that our method allows to solve counting problems with thousands of variables and thousands of constraints. Supportive numerical results are presented.

Session MC04 - Stochastic Networks and Diffusions, Chair: Budhi- raja in CZ 11

Optimal buffer size for a stochastic processing network with a drift

A. P. Ghosh, Statistics Department, Iowa State University, Ames, IA 50011-1210, USA, [email protected] A. Weerasinghe, Mathematics Department, Iowa State University

76 MC04

We consider a one dimensional stochastic control problem that arises in queueing network applications where the controller can choose the arrival and/or service rates and the buffer size b is finite (i.e. customers are rejected if the queue-length exceeds b). This model is similar to the model in Ata-Harrison-Shepp (Annals of Applied Probability, 2005) where the buffer size is kept fixed. We take a long run average cost (“ergodic cost") criterion, which has components corresponding to the cost for arrival/service rate control, holding cost for the customers in the queue as well as a penalty for rejecting customers when the buffer is full. For this cost criterion, we obtain the optimal drift rate (i.e. optimal rates for arrivals and services) as well as the optimal buffer size b*. Our approach involves analyzing a family (indexed by the buffer size b) of ordinary differential equations, where the b-th equation is the Hamilton-Jacobi-Bellman (HJB) equation for the drift control problem in stochastic processing networks with fixed buffer size b. We demonstrate a simple approach to construct solution for this HJB equation (and thus the optimal drift rate) for each buffer size b. This analysis simplifies some of the proofs in Ata-Harrison-Shepp. We make use of some specific characteristics of the family of solutions to choose the optimal buffer size b*.

Characterizations of the invariant measure for a class of reflected diffusions via the extended Skorokhod map

W. Kang, Carnegie Mellon University, U.S.A., [email protected] K. Ramanan, Carnegie Mellon University, U.S.A., [email protected]

In this talk we focus on a class of reflected diffusions in polyhedrons defined via the extended Skorokhod map. We show that the reflected diffusion has certain boundary property, i.e., the pushing process in the reflected diffusion charges zero on the amount time the reflected diffusion spends at the boundary away from a “bad” set V where two or more faces meet. Further, under suitable conditions, the reflected diffusion has an unique invariant measure and such an invariant measure satisfies a basic adjoint relationship.

Diffusion limit for dynamic routing with random service rates

R. Atar, Technion, Israel, [email protected]

The diffusion approach to analyze queueing models has been very successful in providing approximate solutions to complicated problems. In the case of many server limits, however, the ‘simplified’ diffusion models themselves are often hard. I will derive a diffusion limit for a model with a single buffer, many servers, whose service rates are random, and natural assumptions on the routing discipline (i. maximum throughput, ii. fair distribution of idle time among servers), in the form of a particularly simple, one dimensional diffusion with random drift.

Convergence of Invariant Measures for Generalized Jackson Networks

Amarjit Budhiraja, Department of Statistics, University of North Carolina, [email protected] Chihoon Lee, Department of Statistics, University of North Carolina,[email protected]

In a recent work Gamarnik and Zeevi(2006) established that, under certain exponential moment conditions on the network primitives, stationary distributions of suitably scaled critically loaded generalized Jackson networks converge to the unique stationary distribution of a related reflected Brownian motion model. In this work we give a simple proof of the same result without making any exponential moment assumptions. Our moment requirements on the model are precisely those needed for the validity of a heavy traffic central limit theorem.

77 11 – Abstracts

Session MC05 - Flow-Level Models, Chair: Prabhakar in CZ 12

Using the Cavity Method to Analyze Randomized Load Balancing Algorithms with General Flow-size Distributions

Y. Lu B. Prabhakar, Stanford University, [email protected]

Randomized load balancing schemes have been analyzed using the so-called supermarket model: Jobs arrive according to a rate nλ Poisson process at a bank of n rate 1, independent, exponential server queues. An arriving job joins the shortest of d randomly chosen queues. Mitzenmacher (1996) and Vvedenskaya, Dubroshin and Karpelevich (1996) have shown that there is an exponential improvement in the tail of the queue-sizes when going from d = 1 to d > 1 in the above model. This demonstrated the idea that a small amount of choice dramatically improves the performance of load balancing. However, their method of analysis relies on the Poisson arrival and exponential server assumption and doesn’t generalize. In this talk we show how the Cavity Method of Statistical Physics could be used to analyze randomized load balancing in more general settings as well as under changes to the basic model.

Stability of Randomized Load Balancing Algorithms with General Flow-Size Distri- butions

M. Bramson, University of Minnesota, [email protected]

In this talk, we consider randomized load balancing schemes where an arriving job joins the shortest of d randomly chosen queues. As summarized in the related talk by B. Prabhakar, works by Mitzenmacher (1996) and Dubroshin et al. (1996) consider the case with Poisson input and exponentially distributed service times. Here, we consider the stability of generalizations of such systems with arbitrarily distributed interarrival and service times. We discuss results that show that when such systems are subcritical, they are stable for a large family of disciplines and distributions, but that unstable examples exist.

Proportional fairness: stability and insensitivity properties

Laurent Massoulié, Thomson, Paris Research Lab, [email protected]

In this talk a novel characterization of the proportional fairness bandwidth sharing criterion is proposed. Dynamics of concurrent data transfers in a network with such fair sharing are then considered. Using the novel characterization, we show that the corresponding Markov process is in some sense approximately reversible, and use this to extend previously known ergodicity results to the case of phase-type service time distributions.

Stochastic Stability Under Network Utility Maximization: General File Size Distri- bution

M. Chiang Princeton University, [email protected] A. Tang California Institute of Technology, [email protected] D. Shah, Massachusetts Institute of Technology, [email protected]

We prove the stochastic stability of resource allocation under Network Utility Maximization (NUM) under general arrival process and file size distribution with bounded support, for α-fair utilities with α sufficiently

78 MC06 small and possibly different for different sources’ utility functions. In addition, our results imply that the system operating under α-fair utility is 1/(1 + α)-approximate stable for any α ∈ (0, ∞) for any file size distribution with bounded support. Our results are in contrast to the recent stability result of Bramson (2005) for max-min fair (i.e. α = ∞) under general arrival process and file size distribution, and that of Massoulie (2006) for proportional fair (i.e. α = 1) under Poisson arrival process and phase- type distributions. To obtain our results, we develop an appropriate Lyapunov function for the fluid model established by Gromoll and Williams (2006).

Session MC06 - Markov Processes, Chair: Van Doorn in CZ 13

Level-crossing ordering of semi-Markov jump processes and applications to Whittle Networks

F. Ferreira, CEMAT/Universidade de Trás-os-Montes e Alto Douro, Vila Real, Portugal, [email protected] A. Pacheco, CEMAT/Instituto Superior Técnico, Lisboa, Portugal, [email protected]

In this talk we address the level-crossing order, proposed by A. Irle and J. Gani in 2001, which is suited to deal with scenarios in which we are interested on comparing random amounts of time stochastic processes need to exceed certain levels. This stochastic order was investigated by A. Irle and J. Gani for skip-free to the right discrete-time Markov chains (DTMCs), by A. Irle for skip-free to the right semi-Markov processes (SMPs) and continuous-time Markov chains (CTMCs), and by the authors for general DTMCs, SMPs and CTMCs, defined on state spaces order-isomorphic to some bounded or unbounded interval of integers. We investigate the level-crossing ordering of Markov and semi-Markov jump processes defined on common partially ordered Polish spaces, and derive specific results for Whittle networks.

Average optimality for risk-sensitive control with general state space

A. Jaskiewicz´ , Wrocław University of Technology, Institute of Math. and Comp. Sci., 50-370 Wrocław, Poland, [email protected]

This talk concerns discrete-time Markov control processes on a general state space. A one-step cost func- tion is nonnegative and possibly unbounded. The decision maker is supposed to be risk-averse with the constant risk coefficient γ > 0. The risk-sensitive average cost criterion is used as a performance mea- sure. We establish the optimality inequality for risk-sensitive dynamic programming and derive an optimal stationary policy. The result is proved under two different sets of compactness-continuity assumptions, namely, for Markov control processes with weakly continuous transition probabilities (Condition (W)) as well as transition probabilities that are continuous with respect to setwise convergence (Condition (S)). A similar problem for risk-neutral stochastic control models has been examined by Schal (Math. Oper. Res., 1993) using the vanishing discount factor approach. This talk is based on the paper that appears in the Ann. Appl. Probab.

Stochastic binary relations of random variables and processes

L. Leskelä, Centrum voor Wiskunde en Informatica, The Netherlands, lasse.leskela@iki.fi

Stochastic ordering is a fundamental technique for approximating performance quantities of random sys- tems that are too complex to be analyzed exactly. In this talk I will sketch a definition of a stochastic binary relation that extends the notion of stochastic partial order into a relation between probability measures over

79 11 – Abstracts two arbitrary state spaces. An application of Strassen’s coupling theorem then yields easily verifiable nec- essary and sufficient conditions for two Markov processes to preserve a given stochastic binary relation. These conditions turn out useful in approximating stationary distributions of vector-valued random walks and multiclass queueing systems.

Survival in a quasi-death process

E.A. van Doorn, University of Twente, The Netherlands, [email protected]

We consider Markov chains in continuous time with an absorbing coffin state and a finite set S of transient states. When S is irreducible the limiting distribution of the chain as t → ∞, conditional on survival up to time t, is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which S may be reducible, and present a complete solution when the eigenvalue with maximal real part of the generator of the (sub)Markov chain on S has multiplicity one. The application of this result to pure death processes and, more generally, to quasi-death processes will be described.

Session MC07 - Control, Chair: Lelarge in CZ 14

A probabilistic analysis of a leader election algorithm

H. Mohamed, INRIA Rocquencourt, France, [email protected]

A leader election algorithm is an elimination process that divides recursively into tow subgroups an initial group of n items, eliminates one subgroup and continues the procedure until a subgroup is of size 1; the leader. In this paper the biased case is analyzed. We are interested in the cost of the algorithm, i.e. the number of operations needed until the algorithm stops. Using a probabilistic approach, the asymptotic behavior of the algorithm is shown to be related to the behavior of a hitting time of two random sequences on [0, 1].

Investigation of discrete stochastic systems with dynamic distributed routing and delayed information

N.V. Yudaeva, Saratov State University, Russia, [email protected]

Discrete stochastic systems with network structure and dynamic distributed routing of some objects are investigated. Some elements of system make decisions about routing of processed objects. These decisions are based on delayed information about current states of neighboring elements. We consider two models of set of elements, one of which is decision making element. This set is called standard segment of discrete stochastic system. In the first model we take account of processes of forming and processing of information about states of neighboring elements. In the second model we leave out of account of these processes and suppose that this information is always available for decision making element. Queueing networks with dynamic distributed routing are used as models of segment. Expressions for steady-state characteristics of considered systems with and without taking account of delay of information are derived.

On asymptotic optimality of controls for time-varying queues

M. Cudinaˇ , University of Texas at Austin, U.S.A., [email protected] K. Ramanan, Carnegie Mellon University, U.S.A., [email protected] S. Shreve, Carnegie Mellon University, U.S.A., [email protected]

80 MC08

We introduce a general framework for asymptotic optimality of controls for time-varying queues, motivated by applications in manufacturing systems and telecommunications. In particular, we identify controls that are fluid-optimal with respect to a class of performance measures and proceed to discuss the appropriate conditions on the performance measure which result in asymptotic optimality of fluid-optimal controls in the uniform acceleration regime. This analysis is conducted in the settings of a single station and of a tandem queue and uses the framework of strong approximations.

Heavy traffic limit for online bandwidth packing

M. Lelarge, INRIA-ENS, France, [email protected]

We consider the following stochastic bin packing process: the items arrive continuously over time to a server and are packed into bins of unit size according to an online algorithm. The unpacked items form a queue. The items have random sizes with symmetric distribution. Our first contribution identifies some monotonicity properties of the queueing system that allow to derive bounds on the queue size for First Fit and Best Fit algorithms. As a direct application, we show how to compute the stability region under very general conditions on the input process. Our second contribution is a study of the queueing system under heavy load. We show how the monotonic- ity properties allow to derive bounds for the speed at which the stationary queue length tends to infinity when the load approaches one. In the case of Best Fit, these bounds are tight and show that the average  log(1/(1−ρ))  queue size scales as 2 1−ρ when the load ρ approaches one. Our analysis shows connections be- tween our dynamic model, average-case results on the classical bin packing problem and planar matching problems.

Session MC08 - Queueing Models 1, Chair: Fralix in CZ 15

The conditional distribution of the residual service time in the Mn/G/1 queue

Yoav Kerner, The Hebrew university of Jerusalem, [email protected]

We observe an M/G/1 queue with the following variation: an arriving customer is rejected with probability which depends on the queue length. The waiting time of an arriving customer given the queue length is the sum of the service times required by those who stand in line, plus so residual service time of the one who is being served. As the distribution of the service time is given, we are after the conditional distribution of the residual service time given the queue length. This distribution is a function of the rejection probabilities. We give a recursive formula for this distribution and for the corresponding Laplace-Stieltjes transform and mean value. We also give an application in decision making in queues.

Transient Solution to the Time Dependent Single Server Queue with Balking

R.O. Al-Seedy, Department of Mathematics, Faculty of Science, Menoufia University, Menoufia, Egypt A.A. El-Sherbiny, Department of Mathematics, Faculty of Science, Menoufia University, Menoufia, Egypt S.A. El-Shehawy, Department of Mathematics, Faculty of Science, Menoufia University, Menoufia, Egypt S.I. Ammar, Department of Mathematics, Faculty of Science, Menoufia University, Menoufia, Egypt, [email protected]

The transient probabilities for single server queue with balking are derived in a form of an integral equation, when the arrival and departure rates are allowed to vary with time. The used method depends on the

81 11 – Abstracts generating functions technique. In the current paper, the authors show that the transient probabilities for single server queue with balking are satisfying a Volterra integral equation of the second kind.

Relating time- and customer- averages for queues using “forward” coupling from the past

Erol A. Peköz, Department of Operations and Technology Management, Boston University, 595 Commonwealth Avenue, Boston, MA 02215, [email protected] Sheldon Ross, [email protected]

We give a new method for simulating the time-average steady-state distribution of a continuous-time queu- ing system, by extending a “read-once” or “forward” version of the coupling from the past (CFTP) al- gorithm developed for discrete-time Markov chains. We then use this to give a new proof of the “Pois- son Arrivals See Time Averages” (PASTA) property, and a new proof for why renewal arrivals see either stochastically smaller or larger congestion than the time average if inter-arrival times are respectively new better than used in expectation (NBUE) or new worse than used in expectation (NWUE).

A Time-Dependent View of ASTA and Little’s Law

B. H. Fralix, EURANDOM, Eindhoven, The Netherlands, [email protected] R. F. Serfozo, Georgia Institute of Technology, Atlanta, GA, USA, [email protected] G. Riaño, Universidad de Los Andes, Bogotá, Colombia, [email protected]

It is well known that many long-run characteristics of queueing systems (e.g. Little’s law and PASTA) can be proved through the use of Palm probabilities induced by stationary systems. In this talk, we discuss how Palm theory can be used to show such phenomena can also more or less exist in queues before equilibrium is reached. We begin by showing that, in the absence of stationarity, there is still an interesting connection between the stochastic intensity of a point process and the Palm measures induced by this process. This is used to provide necessary and sufficient conditions for a time-dependent version of “ASTA” (our re- sults clearly do not involve time averages) that are very similar in structure to the well-known conditions established by other authors in the stationary case. Consequences of this result include transient versions of PASTA, conditional PASTA and Anti-PASTA. We’ll also discuss how Palm measures can be used to quickly derive time-dependent versions of Little’s law. In particular, the theory can be used to relate any moment of the queue-length at a time t to waiting times spent by customers in the system.

Session MC09 - Inventory 2, Chair: Yao in CZ 16

An EOQ model with state dependence, random yield, and static pricing

Opher Baron, Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6. [email protected] Oded Berman, Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6. [email protected] David Perry, Department of Statistics, University of Haifa. Haifa, Israel 31905. [email protected]

We consider EOQ models with state dependent demand, where the objective is to choose the optimal reorder point and order size to minimize the total inventory cost. We use level crossing methodology to derive the holding cost rate. This approach simplifies significantly the analysis of these well studied models and allows including more realistic demand models, random yield, and pricing. In the latter case our objective is to maximize profits.

82 MC09

Periodic review inventory models

M. Bijvank, VU University Amsterdam, The Netherlands, [email protected] I.F.A. Vis, VU University Amsterdam, The Netherlands, [email protected] G. Koole, VU University Amsterdam, The Netherlands, [email protected]

In the current world of high service levels and tight delivery schedules companies need to adjust their actions to deal with smaller orders that are made more frequently and shorter lead times. Existing inventory models are not designed to deal with these characteristics. Therefore, in this research we consider periodic review models in which lead times are smaller than the length of a review period, i.e., at most one order outstanding. We model the inventory system as a Markov chain and present an approach to analyze the performance of a certain replenishment policy. Based on this an optimal policy can be determined. The approach is general in the sense that it can be used for models with a cost objective or have a service level constraint, but also situations with backordering or lost sales can be considered.

Asymptotically Optimal Solutions to a Joint Replenishment-Transshipment Model

David D. Yao, IEOR Department, Columbia University, New York, NY 10027, USA, [email protected] Yongbo Xiao, School of Economics and Management, Tsinghua University, Beijing, China

We study a model of two stores supplying seasonal demands allowing lateral transshipment. There is a finite planning horizon with a one-time replenishment opportunity before the season starts. While the optimal transshipment policy can be derived by dynamic programming (DP), to optimize the replenishment quantities on top of the DP value function results in a rather intractable problem. We construct an upper bound to the original problem, which takes the form of a concave optimization with a readily derived solution to replenishment quantities. We also propose a simple reservation policy for transshipment, leading to a lower bound. Furthermore, we show that following the replenishment quantities from the upper-bound problem and the reservation policy in the lower bound problem is asymptotically optimal, as the demand rates increase.

Study on inventory management with existence of electronic markets

Ming Xie, IBM China Research Laboratory,[email protected] Diamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, China Wenjun Yin, IBM China Research Laboratory,[email protected] Diamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, China Bin Zhang, IBM China Research Laboratory,[email protected] Diamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, China Xinxin Bai, IBM China Research Laboratory, [email protected] Diamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, China Jin Dong, IBM China Research Laboratory, [email protected] Diamond Building, ZGC Software Part 19, Dong Beiwang road, ShangDi, Beijing 100094, China

In the past several years, Electronic Markets (EM) has been developing rapidly and became an important channel for firms to distribute their products. In this paper we consider a classical single period inventory control problem of an in-store retailer who replenishes its stock from a supplier and sells it to the customer. The retailer jointly decides its ordering quantity and selling price to maximize its profit. Surplus inventory is salvaged to the electronic markets with a dynamic e-price. Demand is determined by both the in-store selling price and the e-price. Excess demand can be satisfied by replenishing inventory from electronic markets. To study this inventory control problem we build a simulation model. By which we investigate inventory policies and pricing strategies under different scenarios with and without electronic markets. The results provide valuable insights for inventory management with existence of electronic markets.

83 11 – Abstracts

Tuesday 8:30am - 10:00am

Session TA01 - Financial Engineering 3, Chair: Kou in CZ 4

Pricing, hedging and calibrating credit from the top down

Kay Giesecke, Management Science & Engineering, Stanford University, USA, [email protected]

A credit derivative is a contingent claim on the aggregate financial loss in a portfolio of credit-sensitive instruments such as loans, bonds or credit swaps. We summarize our recent results on the pricing, hedging and calibration of credit derivatives using point process methods.

Malliavin Greeks without Malliavin calculus

Nan Chen, the Department of System Engineering and Engineering Management, the Chinese University of Hong Kong, Hong Kong, [email protected] Paul Glasserman, the Columbia Business School, Columbia University, New York, USA, [email protected]

We derive and analyze Monte Carlo estimators of price sensitivities (“Greeks”) for contingent claims priced in a diffusion model. There have traditionally been two categories of methods for estimating sensitivities: methods that differentiate paths and methods that differentiate densities. A more recent line of work de- rives estimators through Malliavin calculus. The purpose of this article is to investigate connections be- tween Malliavin estimators and the more traditional and elementary pathwise method and likelihood ratio method. Malliavin estimators have been derived directly for diffusion processes, but implementation typi- cally requires simulation of a discrete-time approximation. This raises the question of whether one should discretize first and then differentiate, or differentiate first and then discretize. We show that in several important cases the first route leads to the same estimators found through Malliavin calculus, but using only elementary techniques. Time-averaging of multiple estimators emerges as a key feature in achieving convergence to the continuous-time limit.

To be determined

Rama Cont, CMAP Department, Ecole Polytechnique, IEOR Department, Columbia University to be determined

Pricing Asian option via double Laplace transform

Ning Cai, Department of IEOR, Columbia University, USA, [email protected] S.G. Kou, Department of IEOR, Columbia University, USA, [email protected]

In this paper, we present an analytic solution for double Laplace transform of the price of a continuous Asian option under the Black-Scholes model, which can be inverted numerically via two-sided Euler inver- sion algorithm. Error bound of the numerical inversion is derived theoretically. Numerical results indicate that this method is efficient and accurate. In addition, a theoretical contribution of the paper is that we give a direct proof of our result by solving a PDE rather than using Bessel process and Lamperti’s relation.

84 TA02

Session TA02 - Stochastic Geometry, Chair: Van Lieshout in CZ 5

Wiener sausage and sensor networks

E. Spodarev, Ulm University, Germany, [email protected] R. Cernýˇ , Charles University, Prague, Czech Republic, [email protected] S. Funken, Ulm University, Germany, [email protected] D. Meschenmoser, Ulm University, Germany, [email protected] J. Rataj, Charles University, Prague, Czech Republic, [email protected] V. Schmidt, Ulm University, Germany, [email protected]

The parallel r-neighborhood of a path of the Brownian motion in d dimensions is called after Norbert Wiener a Wiener sausage. This random compact set can serve as a model for the coverage area of a Brownian sensor able to detect a target within reach r. In this sense, Boolean models of Wiener sausages can be used to describe the behavior of sensor networks trying to detect an immobile intruder. In this talk, some properties of (Boolean models of) Wiener sausages are studied. In particular, the capacity functional, the volume fraction, the specific surface area, the convariance function and the spherical contact distribution function are calculated. Approximations of the Wiener sausage by polyconvex sets and the convergence of the corresponding curvature measures are addressed as well.

Maximum likelihood estimation for random sequential adsorption

M.N.M. van Lieshout,CWI,P.O. Box 94079, 1090 GB Amsterdam, The Netherlands,EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Packing problems occur in many physical and biological processes. Hard ‘particles’ such as cars, monomers, or proteins, arrive in some bounded Euclidean window according to a constant rate Poisson process, and select a position for themselves according to some fixed distribution. If there would be no overlap with an established particle, the new one attaches itself to the selected location, otherwise it leaves the system. The process continues for a given time, or until there is no room for another particle to establish itself without intersecting existing ones. To date, most attention has focused on probabilistic work rather than inference. In this talk, we take the statistical viewpoint. We place the RSA model in the context of Markov sequential spatial processes, derive its Radon–Nikodym derivative, and consider maximum likelihood estimation of its parameters. Note that maximum likelihood is especially convenient in exponential family models, but that RSA exhibits a highly non-linear dependence. This research is supported by the Technology Foundation STW, applied science division of NWO, and the technology programme of the Ministry of Economic Affairs (project CWI.6156 ‘Markov sequential point processes for image analysis and statistical physics’). Reference: M.N.M. van Lieshout Maximum likelihood estimation for random sequential adsorption. Ad- vances in Applied Probability (SGSA) 38:889–898, December 2006.

Random field model for fingerprint features

Elke Thönnes, Department of Statistics and Centre for Scientific Computing, University of Warwick, Coventry, CV4 7AL, UK, [email protected]

Fingerprint experts classify information from human fingerprints into three categories: level one describes the overall pattern of skin ridges; level two concerns singularities within the ridge flow such as ridge endings or bifurcations; and finally, level three consists of high level features such as shape of ridge endings or the location and shape of sweat pores.

85 11 – Abstracts

Sweat pores only occur on ridges and so there is a very intricate relationship between level two and level three features. In this talk I will discuss joint work with Mr J. Su and Prof. W.S. Kendall which represents first steps at modelling the ridge structure conditional on observed sweat pore locations. Using a non-linear transformations on the pore pattern, a random tensor field is defined that indicates principal flow directions. The field can then be used to locate singularities within the ridge flow or to classify pores according to ridge membership.

Opportunistic Routing on Signal-to-Interference-and-Noise Ratio Graphs

F. Baccelli, ENS/INRIA Paris, France, [email protected] B. Błaszczyszyn, ENS/INRIA Paris, France and University of Wrocław, Poland, [email protected] P. Mühlethaler, INRIA Rocquencourt, France, [email protected]

In opportunistic routing in multi-hop mobile wireless networks, for each packet and each hop, one dy- namically selects as next relay the node that captures the packet transmission and which is the nearest to the destination. Such a scheme allows each packet to take advantage of the local pattern of transmissions and at any slot and at any hop. Simulations show that such routing schemes significantly improve the per- formance of multi-hop networks compared to conventional routing algorithms, where packets are routed on a pre-defined route obtained by a shortest path routing protocol. We propose a stochastic-geometry framework based on point maps to explain some observations made by simulation. In particular, we prove that our opportunistic routing algorithms can be optimized so as to minimize the average end-to-end delay incurred by a packet over long paths and that the optimum setting does not depend on the node density in the homogeneous Poisson case. Various challenging problems remain open.

Session TA03 - Rare Event Simulation 2, Chair: Juneja in CZ 10

Alternative efficiency proof and interpretations for a recent state-dependent impor- tance sampling scheme

P.T. de Boer, University of Twente, The Netherlands, [email protected] W.R.W. Scheinhardt, University of Twente, The Netherlands, [email protected]

Recently, a state-dependent change of measure for simulating overflows in the two-node tandem queue has been proposed by Dupuis, Sezer and Wang (2005). They derive it using game-theory, and prove its asymptotic efficiency. We present a simpler efficiency proof, by decomposing the likelihood ratio of a sample path into a factor that only depends on the initial and final states, and a factor whose influence can be shown to be bounded. Furthermore, we give intuitive interpretations for several of the abstract quantities involved in deriving this change of measure. In particular, we show that the conditions derived using game theory are equivalent to well-known conditions on the likelihood-ratio of cyclic paths.

Importance sampling for heavy-tailed random walk

Achal Bassamboo, Northwestern University, Evanston, IL, USA, [email protected] Sandeep Juneja, Tata Institute of Fundamental Research, Mumbai, India, [email protected] Assaf Zeevi, Columbia University, NY, NY, USA, [email protected]

In this talk we consider a negative drift random walk with heavy-tailed increments. We are interested in estimating the probability of this random walk exceeding a a large threshold starting from zero before

86 TA04 taking a negative value, using simulation. We use Importance sampling technique for reducing the variance of this estimator. Earlier results have shown that using a state-independent change-of-measure one cannot obtain an efficient measure. We present an asymptotically efficient state-dependent change-of-measure for estimating this probability.

A Reduced-Form Model With Short-Time Contagion For Credit Risk

Jose Blanchet, [email protected] Li Zhan, [email protected]

Standard intensity default models do not capture contagion effects (basically the possibility of multiple simultaneous defaults). We discuss a reduced form model that captures contagion effects in short periods of time (a feature that can be useful to model “panic” situations). The proposed model has convenient analytic features and induces loss distributions that possess heavier tails than standard intensity models. We also discuss associated rare-event simulation methodology for estimating the tail of the loss distribution.

Minimizing Tail Probabilities: Asymptotic Analysis and Efficient Simulation Opti- mization

Sandeep Juneja, Tata Institute of Fundamental Research, India, [email protected] Himanshu Kalra, Tata Institute of Fundamental Research, India

We consider a resource allocation problem where the objective is a tail probability associated with max- imum of sums of a few random variables. Applications include stochastic activity networks where our interest is in large project delays, and portfolio optimization where we are concerned about the probability of large losses. We first consider a suitably scaled asymptotic of the tail probability as an approximate objective function. The resulting approximate problem is a convex programming problem and is easily solved providing insight into the nature of the solution to the original problem. We then solve the exact problem using simulation optimization. Here we introduce a change of variable technique that allows a single importance sampling probability measure to asymptotically efficiently estimate the objective over the entire allocation space. We see that this considerably speeds up determining the optimal solution via simulation particularly when insights from the asymptotic solution are utilized.

Session TA04 - Scheduling and Stability, Chair: Hasenbein in CZ 11

Fluid and Diffusion Limits of a Multiclass Overloaded Abandonment Queue under Priority Scheduling

O. Jennings, [email protected] Josh E. Reed

Consider a sequence of overloaded multiclass queues with abandonment, indexed by n. The customer classes are split into priority subgroups and, within each subgroup, jobs are processed in a FIFO fashion. The sequence of arrival and service rates are linear in n, whereas the abandonment distribution is constant. All r.v.’s have general distributions. In the limit, only one of the subclasses has a nondegenerate waiting time value. For this subclass, the limiting diffusion-scaled, abandonment-filtered virtual waiting time pro- cess is an OU process whose drift is proportional to the traffic intensity-weighted sum of the associated abandonment densities, each evaluated at the limiting deterministic waiting time value.

87 11 – Abstracts

Stability of Non-random Critically Loaded Networks under MaxWeight Scheduling

D.M. Andrews, Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USA, [email protected] K. Jung, MIT, Cambrdge, MA, [email protected] A.L. Stolyar, Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USA, [email protected]

We study a queueing network, operating in discrete time, with deterministic (periodic) exogenous arrival processes and fixed service rates at the nodes. The MaxWeight algorithm is used for the combined routing and scheduling. (Thus, the network evolution is deterministic.) We consider the case when network is exactly critically loaded, and prove that all queues remain bounded. The result has implications for the running time of the MaxWeight algorithm, when it is used to solve multicommodity flow problems. In particular, for a fixed problem instance, the algorithm can be applied to find a flow that is within a (1 − )-factor of optimal in time O(1/) (improving the previous bound of O(1/2)).

Undecidability results in the theory of queueing networks and Skorokhod problem

D. Bertsimas, MIT, USA, [email protected] D. Gamarnik, MIT, USA, [email protected] D. Katz, MIT, USA, [email protected]

We consider a deterministic multiclass queueing network with a mixture of finite and infinite buffer capac- ities, operating under a static buffer priority scheduling rule and investigate the question of stability of such class of policies. To the day no tight conditions for stability of such policies are known. We establish that stability is an algorithmically undecidable property, thus no constructive tight characterization is possible. We further extend this result to a deterministic version of a Skorokhod mapping problem, establishing a similar undecidability result.

Networks with Parameter Uncertainty: Stochastic Fluid Models with Recourse

John J. Hasenbein, Graduate Program in Operations Research and Industrial Engineering, University of Texas at Austin, Austin, Texas, USA, [email protected] Burak Buke, David P. Morton, Graduate Program in Operations Research and Industrial Engineering, University of Texas at Austin, Austin, Texas, USA {bukeb,morton}@mail.utexas.edu

We consider fluid models in which there is uncertainty in the parameters defining the networks. We inves- tigate the problem of minimizing makespan when a stochastic programming decision structure is imposed on the control mechanisms. The controller must make a control decision before having full knowledge of the parameters, and she may also update that decision after gaining some knowledge. The fluid model problem is in general not analytically tractable, and we present algorithms to numerically solve the problem efficiently.

Session TA05 - Ad Hoc Networks, Chair: Van den Berg in CZ 12

Efficient Resource Control for the EDCA and HCCA Mechanisms in Multi-Rate IEEE 802.11e Networks

Vasilios A. Siris, Inst. of Comp. Science, FORTH and Dept. of Comp. Science, Univ. of Crete, [email protected] Costas Courcoubetis, Department of Informatics, Athens Univ. of Economics and Business, [email protected]

88 TA05

We investigate the problem of efficient resource control for elastic traffic over the EDCA (Enhanced Dis- tributed Channel Access) and HCCA (Hybrid Coordination Function - HCF - Controlled Channel Access) mechanisms of IEEE 802.11e. Our approach considers an economic modelling framework based on con- gestion pricing that captures how various factors, such as the probability of attempting to transmit a frame, the use of the basic CSMA/CA or the RTS/CTS procedure, the transmission opportunity (TXOP), and the physical layer transmission rate, contribute to congestion. Additionally, we consider the joint control of the EDCA and HCCA mechanisms, which allows us to determine the optimal sharing of the wireless channel between the two access mechanisms. We discuss the application of the framework for achieving class- based throughput differentiation, for performing explicit congestion notification (ECN) marking based on the level of congestion in the wireless channel, and for modelling the performance of TCP congestion control over EDCA. In these application scenarios we discuss how to estimate the optimum minimum con- tention window and the congestion prices based on the 802.11e parameters and actual measurements, in order to achieve efficient channel utilization.

Performance Modeling of a Bottleneck Node in an IEEE 802.11 Ad-hoc Network

Frank Roijers, TNO Information and Communication Technology, the Netherlands, [email protected] Hans van den Berg, TNO Information and Communication Technology, the Netherlands, [email protected] Michel Mandjes, Korteweg-de Vries Institute, University of Amsterdam, the Netherlands, [email protected]

The IEEE 802.11 MAC-protocol, often used in ad-hoc networks, has the tendency to share the capacity equally amongst the active nodes, irrespective of their loads. An inherent drawback of this fair-sharing pol- icy is that a node that serves as a relay-node for multiple flows is likely to become a bottleneck. This paper proposes a flow-level performance model of such a bottleneck node using fluid-flow analysis. Assuming Poisson initiations of new flow transfers at the bottleneck node, we obtain insightful, robust, and explicit expressions for characteristics related to the overall flow transfer time, the buffer occupancy, and the packet delay at the bottleneck node. The analysis is enabled by a translation of the behavior of the bottleneck node and the source nodes in terms of an M/G/1 queueing model. We conclude the paper by an assessment of the impact of alternative capacity sharing amongst source nodes and the bottleneck in order to improve the performance of the bottleneck.

Collaboration and incentives in wireless ad-hoc networks

R.J. Gibbens, University of Cambridge, United Kingdom, [email protected] P.B. Key, Microsoft Research Cambridge, United Kingdom, [email protected]

This talk considers how incentives for collaboration can be incorporated into a simplified model of com- munication in wireless ad-hoc networks. Such networks comprise heterogeneous, possibly mobile, sources and resources which are coupled through a common distributed wireless communication channel. We ex- amine some of the optimization and pricing methods proposed to coordinate jointly beneficial behaviour of the network and explore some of their game-theoretic consequences.

A fluid model for a relay node in an ad-hoc network: the case of heavy-tailed input

R. Bekker, Vrije Universiteit Amsterdam, The Netherlands, [email protected] M. Mandjes, Korteweg-de Vries Institute for Mathematics, The Netherlands, [email protected]

Relay nodes in an ad-hoc network can be modelled as fluid queues, in which the available service capacity is shared by the input and output. In this paper such a relay node is considered; jobs arrive according to a Poisson process and bring along a random amount of work. The total transmission capacity is fairly shared,

89 11 – Abstracts meaning that, when n jobs are present, each job transmits traffic into the queue at rate 1/(n + 1) while the queue is drained at the same rate of 1/(n + 1). Where previous studies mainly concentrated on the case of exponentially distributed job sizes, the present paper analyzes the case of regularly varying jobs. The focus lies on the tail asymptotics of the sojourn time S. Using sample-path arguments, it is proven that P {S > x} behaves roughly as the residual job size, i.e., if the job sizes are regularly varying of index −ν, the tail of S is regularly varying of index 1 − ν. The tail asymptotics of other performance metrics, such as the workload in the queue, the flow transfer time and the queueing delay, are also addressed.

Session TA06 - Call Centers 2, Chair: Koole in CZ 13

Shift scheduling for heterogeneous contact centers via mixed-integer programming

S. Helber, Leibniz University Hannover, Germany, [email protected] K. Henken, [email protected]

We present a discrete-time decision model to solve the shift-scheduling problem for a given day in a large inbound contact center which uses different agent classes to serve different classes of impatient customers with retrials. The dynamics of the contact center due to time-varying arrival rates and time dependent staffing levels are modeled via a fluid model. Mixed-integer programming is used to approximate profit maximizing staffing levels over the day. The results are compared to simulation.

A real-time routing policy for call centers

S.A. Pot, Vrije Universiteit of Amsterdam, The Netherlands, [email protected]

We consider a multi-skill call centers, having different job types and several agent groups. A routing policy is presented that deals with unexpected fluctuations in the arrival intensities. We show that the policy yields the best achievable service levels under certain circumstances.

The Hayward Approximation with Queueing Systems

Zeynep Aksin, Koç University, Turkey, [email protected] Jean-Christophe Van den Schrieck, Université catholique de Louvain, Belgium, [email protected]

The Erlang-C formula is widely known in queueing theory. Its utilization is however limited to Poisson arrivals and exponential service time distribution. In this talk we propose to investigate if this formula can be used in the case the arrival flow is an overflow. To be more specific, we want to check whether the Hayward Approximation, which approximates the loss probability in case of overflow arrivals, could be extended to queueing systems.

Staffing Call Centers with Differentiated Levels of Service: Approximate Solutions via Constraint Dualization

Achal Bassamboo, Northwestern University, Evanston, IL, USA. [email protected] Assaf Zeevi, Columbia University, NY, NY, USA. [email protected]

We consider a call center model with multiple customer classes and multiple server pools. Calls arrive randomly over time; customers are impatient and may abandon the system before their service commences.

90 TA07

The objective is to minimize staffing costs subject to some quality of service constraints. We consider two specific examples of the latter: i) an upper bound on the fraction of customers who abandon; and ii) an upper bound on the waiting time of the customer. The approach proposed in this paper has two key ingredients. The first builds on suitably approximating the dynamics of the original system, which in turn simplifies the form of the constraint. The second step uses duality theory to convert this constrained problem into an unconstrained (lagrangian) formulation; the Lagrange multipliers can be interpreted as penalties for constraint violation. This problem can be solved efficiently using saddlepoint algorithms. The resulting prescription for the staffing level (and the corresponding routing control) can be shown to be almost optimal in a suitable mathematical sense, i.e., its performance comes close to the best achievable performance in the original constrained optimization problem.

Session TA07 - Statistics 1, Chair: Segers in CZ 14

Martingale approach on the average run lengths of Exponentially EWMA control charts

Y. Areepong, University of Technology, Sydney, Australia, [email protected] S. Sukparungsee, University of Technology, Sydney, Australia, [email protected] A. Novikov, University of Technology, Sydney, Australia, [email protected]

Approximate computational methods, using the Martingale approach to determine the expected first pas- sage times in Exponentially EWMA charts, are demonstrated and extended to more general heavy-tailed distribution. Explicit formula are presented and compared with Markov Chain Approach and Monte Carlo Simulation. The analytical and numerical results are given to show the efficacy of this approach.

Comparative study of change detection properties of Poisson EWMA control proce- dure

S. Sukparungsee, University of Technology, Sydney, Australia, [email protected] A. Novikov, University of Technology, Sydney, Australia, [email protected]

This article aims to examine the analytical approximations called the ”Martingales Technique” to show how the exact lower bounds of the Average Run Lengths (ARLs) of Poisson Exponentially Weighted Moving Average (PEWMA). Based on this technique, it could be help to reduce the time of finding the ARLs. The effectiveness of this study will be done by comparison of the EWMA properties and CPU timing. The optimal design one-sided and two-sided case are provided as a combination of optimal parameters.

Benchmarking of Fitting Algorithms for Continuous Phase-Type Distributions

J.F. Pérez, Universidad de los Andes, Colombia, [email protected] G. Riaño,Universidad de los Andes, Colombia, [email protected]

Phase-Type distributions have shown to be useful in the stochastic modeling of real systems. Many efforts have been made to provide algorithms to fit the parameters of a Phase-Type distribution from raw data. In this paper, we present a comparative analysis of some recent fitting algorithms. These algorithms are tested against several data sets and their performance is measured in different ways. The results give insight about their behavior and the situations in which one of them can overperform others. The analysis provides information that can be relevant when selecting one of those methods for a particular application.

91 11 – Abstracts

Regularly varying multivariate time series

Bojan Basrak, Zagreb University, Croatia, [email protected] Johan Segers, Université catholique de Louvain, Belgium, [email protected]

Extremes of univariate heavy-tailed first-order Markov chains are known to behave under general condi- tions as a multiplicative random walk, the tail chain. This property is found to be a particular case of a much more general one stating that a stationary multivariate time series is multivariate regularly varying if and only if the following property holds: conditionally on the value of the process at a fixed time point being large (in norm), the process converges in the sense of finite-dimensional distributions to a limit process, the tail process. This tail process is found to possess a number of remarkable properties. Moreover, all kinds of interesting quantities can be expressed in terms of this tail process: limit point processes, multivariate extremal indices, functionals of clusters of multivariate extremes. For general multivariate Markov chains, the tail process takes a special form, yielding neat formulas in certain examples, for instance for solutions to stochastic difference equations or moving averages with random coefficients.

Session TA08 - Polling Models, Chair: Winands in CZ 15

Polling systems with parameter renewals

I. MacPhee, [email protected]

Many widely used models of queueing systems assume time homogeneity for arrival streams and service distributions. This talk will discuss results about the long term behaviour of an exhaustive polling system with arrival rates and service time distribution that are randomly selected each time the server switches to a new queue. We suppose the server visits the stations in some cyclic ordering. Our main result establishes which moments of τ, the time taken to empty the system, are finite. The key parameter is the top Lyapunov exponent of a particular random matrix. It turns out that critical/null recurrent behaviour can occur in a thick region of the parameter space. This is joint work with M. Menshikov, D. Petritis and S. Popov. One paper, A Markov chain model of a polling system with parameter regeneration, is to appear in Annals of Applied Probability and we have recently completed another.

M/G/∞ polling systems with random visit times

Maria Vlasiou, Georgia Institute of Technology, H. Milton Stewart School of Industrial & Systems Engineering, 765 Ferst Drive, Atlanta GA 30332-0205, USA, [email protected] Uri Yechiali, Department of Statistics and Operations Research, School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel, [email protected]

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process and service time requirement of each individual customer has a general pdf. Thus, each of the queues comprising the system is, in isolation, an M/G/∞-type queue. A job that is not completed during a visit will have a new service time requirement sampled from the service time distribution of the corresponding queue. We derive the expected values and probability generating functions of the queue lengths, as well as the mean and Laplace-Stieltjes transform of the sojourn time of a customer. Moreover, we identify the policy that maximises the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is independent of the number of customers present at the various queues at the start of the cycle. In other words (and somewhat surprisingly), additional

92 TA09 information regarding the state of the system at the start of a cycle does not lead to an improvement of the optimal policy.

Polling Systems with Two-Phase Gated Service: Heavy Traffic Results for the Waiting- Time Distributions

R.D. van der Mei, CWI and Vrije Universiteit, The Netherlands, [email protected] J.A.C. Resing, Eindhoven University of Technology, The Netherlands, [email protected]

We consider an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch- over time distributions, and with so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce fairness among the different customer classes. For this model, we use the classical theory of multi-type branching processes (MTBPs) to derive closed-form expressions for the Laplace- Stieltjes Transform (LST) of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic (HT) scalings.

Branching-type polling systems with large setups

E.M.M. Winands, Eindhoven University of Technology, The Netherlands, [email protected]

The present paper considers the class of polling systems that allow a multi-type branching process inter- pretation. This class contains the classical exhaustive and gated policies as special cases. We present an exact asymptotic analysis of the delay distribution in such systems, when the setup times tend to infinity. The motivation to study these setup time asymptotics in polling systems is based on the specific application area of base-stock policies in inventory control. Our analysis provides new and more general insights into the behavior of polling systems with large setup times.

Session TA09 - Switch Scheduling, Chair: Moallemi in CZ 16

Switch scheduling via message passing

M. Bayati, Microsoft Research, USA, [email protected]

Max-product “belief propagation" is an iterative, local, message-passing algorithm for finding the maxi- mum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding, com- puter vision and combinatorial optimization which involve graphs with many cycles, theoretical results about both correctness and convergence of the algorithm are known in few cases. On the other hand finding Maximum Weight Matching (MWM) in a weighted complete bipartite graph is a central problem in designing an input-queued switch for high speed internet routers. In this talk we use max-product to design an iterative and distributed algorithm for finding MWM. Even though the underlying bipartite graph has many short cycles, we find that surprisingly, the max-product algorithm always converges to the correct answer. We also show that for a graph of size n, the computational cost of the algorithm scales as O(n3), which is the same as the computational cost of the best known algorithm. Finally, we establish the precise relation between the max-product algorithm and the celebrated auction algorithm proposed by Bertsekas. This suggests possible connections between dual algorithm and max- product algorithm for discrete optimization problems.

93 11 – Abstracts

How to avoid doing research “10 years ahead of time”

Amin Firoozshahian Vahideh Hosseinkeh Ashish Goel Balaji Prabhakar, Stanford University, [email protected]

A thread of research dating to the mid-90s is concerned with the emulation of an output-queued (OQ) switch with a combined input- and output-queued (CIOQ) switch. It showed that in order to emulate an N port OQ switch under adversarial inputs, it is necessary and sufficient for the CIOQ switch to run at a speedup of 2 − (1/N). This result validated a belief that was prevalent among implementors and researchers. However, the Stable Marriage algorithms developed in this work were considered too complicated to imple- ment, and implementors contented themselves with the knowledge that 2 is “the right amount of speedup” and ignored the algorithms. Recently, a combination of factors has made Stable Marriage algorithms attractive for implementation. This talk narrates the story and gives the technical details.

Packet Election For Fair Scheduling

S. Jagabathula, Massachusetts Institute of Technology, [email protected] V. Doshi, Massachusetts Institute of Technology, [email protected] D. Shah, Massachusetts Institute of Technology, [email protected]

In our previous work (2006), we conjectured that Max-Weight Max-Size Matching (MWMSM) scheduling algorithm is both throughput and (average) delay optimal for an input-queued switch based on insights obtained from the heavy traffic analysis of a class of Max-Weight Matching scheduling algorithms. In this talk, we (almost) prove this conjecture: we show that MWMSM algorithm is a constant factor approxima- tion (with respect to average queue-size) of an optimal scheduling algorithm for any (time) i.i.d. arrival process.

Approximate Dynamic Programming for Switch Scheduling

Ciamac C. Moallemi, Stanford University, USA, [email protected] Sunil Kumar, Stanford University, USA, [email protected] Benjamin Van Roy, Stanford University, USA, [email protected]

We develop an approach based on temporal-difference learning to address scheduling problems in complex queueing networks such as those arising in service, communication, and manufacturing systems. One novel feature is the selection of basis functions, which is motivated by the gross behavior of the system in the asymptotic regime of heavy traffic. Another is the use of polytopic structure to efficiently identify desired actions from an intractable set of alternatives. This metholodogy is applied to input-queued crossbar switch models with up to hundreds of queues and quadrillions of alternative actions. It is shown yield scheduling policies outperforming a heuristic recently shown to have certain optimality properties in the heavy traffic scale. We also extend the approach to a setting where aspects of the queueing network are not modeled and we must rely instead on empirical data. This data-driven approach is useful, for example, when the statistical structure of arrivals is poorly understood but historical data traces are available.

94 TB01

Tuesday 10:30am - 12:00pm

Session TB01 - Finance and Ruin, Chair: Collamore in CZ 4

A Dirichlet bridge sampling of the variance gamma process: Pricing path-dependent options

Laura Ballotta, Faculty of Actuarial Science and Insurance, Cass Business School London, [email protected] Dimitrina Dimitrova, Faculty of Actuarial Science and Insurance, Cass Business School London, [email protected] Vladimir Kaishev, Faculty of Actuarial Science and Insurance, Cass Business School London, [email protected]

A new method for sampling a Variance Gamma (VG) process, called Dirichlet bridge sampling, is proposed and is shown to represent a generalization of the known gamma bridge sampling method. Dirichlet bridge sampling allows immediate generation of the entire trajectory of the process over a certain period of time, avoiding sequential sampling at arbitrary intermediate points in time as it is the case, instead, with the gamma bridge method. We explore the efficiency of the proposed simulation methodology and apply some variance reduction techniques such as stratification. The proposed method is then tested against approaches already existing in the literature, such as sequential sampling and bridge sampling of the VG process. In particular, we price path-dependent options such as Asian options, lookback options and barrier options.

Applications of series representations for Lévy processes

Z. Michna, Wroclaw University of Economics, Poland, [email protected]

We consider α-stable Lévy processes and gamma Lévy process. We use a series representation of the α- stable Lévy process to condition on the largest jump. Then we get a certain decomposition into a simple process and a Lévy process which has some finite exponential moments. We show that this Lévy process converges to the α-stable Lévy process uniformly on compact sets with probability one. In other limit we can approximate this process by Brownian motion. We apply this decomposition of an α-stable Lévy process to study asymptotic behavior of the distribution of supremum for certain α-stable processes. Using the same technique as for α-stable Lévy processes we find an asymptotic behavior of finite time ruin probability for a gamma Lévy process.

Stochastic recurrence equations and ruin in a general Markovian environment

J.F. Collamore, University of Copenhagen, Denmark, [email protected] H. Nyrhinen, University of Helsinki, [email protected].fi

We develop sharp large deviation asymptotics for the probability of ruin in a Markov-dependent stochastic economic environment. Namely, we study the maximal behavior for the system of equations

Wn = B1 + A1 B2 + · · · + (A1 ··· An−1)Bn, n ≥ 1, where (An, Bn) is a function of an underlying Markov chain in general state space; and show   −R P sup Wn > u ∼ Cu as u → ∞. (11.1) n

95 11 – Abstracts

The left-hand side of (1) may be equated to the probability of ruin of an insurance company, given an initial capital of u, with Markov-dependent stochastic investment returns. In addition to (1), we also mention some related results for the study of perpetuities, and for the extremes of the ARCH(1) and GARCH(1,1) financial time series models. Our results build upon work of Goldie (Ann. Appl. Prob., 1991), who has developed tail asymptotics for independent sequences of random variables subject to a random recurrence equation. We adopt a general approach based on the theory Harris recur- rent Markov chains and the associated theory of nonnegative operators, and meanwhile develop certain recurrence properties for these operators.

Managing Inventory of Items with Replacement Warranty

Wei Huang, SAS Institute Inc, Cary, NC 27511 Vidyadhar Kulkarni, Department of Statistics and Operations Research University of North Carolina, Chapel Hill, NC 27599 Jayashankar M. Swaminathan, The Kenan-Flagler Business School University of North Carolina, Chapel Hill, NC 27599

In this paper we study a firm that faces demand from two sources: demand for new products and demand to replace failed items under warranty. We model this setting as a multi-period single product inventory prob- lem where the new demand in different periods are independent and the demand for replacing failed items is dependent on the number and age of items under warranty. We consider backlogging and emergency supply cases, and study both discounted cost and average cost cases. We prove the optimality of the w- dependent base stock ordering policy where the base stock level is a function of w, the vector representing the number of items at different ages currently under warranty. For the special case, where the demand for new products is stationary, we prove the optimality of a stationary w-dependent base stock policy for the finite horizon discounted and the infinite horizon discounted and average cost cases. In our computational study, we find that an optimal integrated policy can lead to 37% average improvement in expected costs when compared to a policy that neglects warranty repairs.

Session TB02 - Random Partitions and Partition-valued Processes, Chair: Gnedin in CZ 5

Small time asymptotics of Lambda-coalescents

J. Berestycki, Université de Provence, [email protected] N. Berestycki, University of British Columbia, [email protected] V. Limic, Université de Provence, [email protected]

Bertoin and Le Gall have recently remarked that there is a simple bijective transformation that maps the measure 3 of a 3-coalescent onto the branching mechanism 9 of a continuum branching process such that Schweinsberg’s criterion on 3 for coming down from infinity in the coalescence corresponds exactly to Grey’s criterion for extinction in finite time for the branching process. However no probabilistic explanation of this fact was known. By shedding more light on the relations (which have been the focus of much attention lately) between 3- coalescent, continuous-state branching processes and so-called generalized Flemming-Viot super-processes, we are able to give such an explanation, and along the way we obtain an almost sure asymptotic formula for the number of blocks at small-times, which until now was known only in very specific cases.

96 TB03

Gaussian fluctuations for Plancherel partitions

L. Bogachev, University of Leeds, UK, [email protected]

The talk is based on joint work with Z. Su (Zhejiang University). We obtain a central limit theorem for Young diagrams under the Plancherel measure in the bulk of the “spectrum” of partitions of large integers n. More specifically, we prove that, under a suitable normalization (growing like p log n ), the corresponding random process converges, in the FDD sense, to a Gaussian process with independent values. We also discuss the link with an earlier result by S. Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on the Poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes.

Ancestral processes in population genetics - exchangeable coalescents

Martin Möhle, University of Düsseldorf, Germany, [email protected]

Coalescents are partition-valued Markovian processes with a block-merging transition mechanism. During each transition blocks merge into single blocks. These processes can arise in the limit as the total population size tends to infinity when studying the genealogy of a sample of individuals from an exchangeable finite population. Coalescents play an important role in the theory of coagulation and fragmentation. They have remarkable relations to measure-valued processes, random recursive trees and other combinatorial objects such as partition structures and sampling formulas. This talk introduces into the theory of coalescents and reviews recent mathematical research and biological applications around these stochastic processes.

Random partitions and the site frequency spectrum in population genetics

Nathanaël Berestycki, University of British Columbia, Canada, [email protected] Julien Berestycki, Université de Provence, France, [email protected] Jason Schweinsberg, University of California at San Diego, USA, [email protected]

Suppose we take a sample of size n from a population and follow the ancestral lines backwards in time until the most recent common ancestor. Under standard assumptions, this process can be approximated by Kingman’s coalescent, in which two lineages merge at rate one. Mutations that have occurred since the time of the most recent common ancestor will lead to segregating sites, which are positions in the DNA at which not all individuals in the sample are the same. If we denote by Mk the number of mutations that affect k individuals in the sample, then the sequence M1,..., Mn−1 is called the site frequency spectrum. We will explain how a result on random partitions of Karlin (1967) allows us to understand how the site frequency spectrum would be affected if some individuals in the population have unusually large numbers of offspring.

Session TB03 - Gradient Estimation, Chair: Heidergott in CZ 10

Policy Gradient Methods in Machine Learning

Jan Peters, University of Southern California, Los Angeles, USA, [email protected] Evangelos Theodorou, University of Southern California, Los Angeles, USA, [email protected] Stefan Schaal, University of Southern California, Los Angeles, USA, [email protected]

97 11 – Abstracts

We present an in-depth survey of policy gradient methods as they are used in the machine learning com- munity for optimizing parameterized, stochastic control policies in Markovian systems with respect to the expected reward. Despite having been developed separately in the reinforcement learning literature, policy gradient methods employ likelihood ratio gradient estimators as also suggested in the stochastic simulation optimization community. It is well-known that this approach to policy gradient estimation traditionally suffers from three drawbacks, i.e., large variance, a strong dependence on baseline functions and a ineffi- cient gradient descent. In this talk, we will present a series of recent results which tackles each of these problems. The variance of the gradient estimation can be reduced significantly through recently introduced techniques such as optimal baselines, compatible function approximations and all-action gradients. How- ever, as even the analytically obtainable policy gradients perform unnaturally slow, it required the step from ‘vanilla’ policy gradient methods towards natural policy gradients in order to overcome the inefficiency of the gradient descent. This development resulted into the Natural Actor-Critic architecture which can be shown to be very efficient in application to motor primitive learning for robotics.

Strong Properties of Weak Derivatives

Haralambie Leahu, Vrije Universiteit, The Netherlands, [email protected] Bernd Heidergott, Vrije Universiteit, The Netherlands, [email protected] Arie Hordijk, Leiden University, The Netherlands, [email protected]

In this talk we provide bounds on cost functions over a finite collection of independent random variables. Our analysis is based on a study of the relation between weak differentiability and Fréchet differentiability. By applying a fundamental result from the theory of Banach spaces, we show that weak differentiability implies norm Lipschitz continuity. This result leads to bounds on the sensitivity of finite products of probability measures in norm sense. We illustrate the application of these results by studying parameter- free perturbation analysis. Our main application is the expected transient waiting time in a G/G/1 queue.

Gradient Estimation for a Multi-Component Maintenance System Using Measure- Valued Differentiation

Taoying Yuan, Vrije Universiteit, The Netherlands, [email protected] Bernd Heidergott, Vrije Universiteit, The Netherlands, [email protected]

We consider a multi-component maintenance system with an age replacement policy: when a component fails, it is replaced immediately, and all components older than a threshold age θ are preventively replaced. With each maintenance action, costs are associated. We derive three unbiased gradient estimators based on the measure-valued differentiation approach for the derivative of the long-run average cost with respect to θ: the phantom estimator, the randomized phantom estimator and the combined phantom estimator. The combined estimator is the first unbiased gradient estimator that is feasible for systems with a large number of components. In addition, a new technique is introduced that allows to control the number of phantoms to be simulated in parallel for obtaining the gradient estimator, and a novel approach for efficiently implementing the phantom generation is presented.

Measure-Valued Differentiation for Random Horizon Experiments

B. Heidergott, Vrije Universiteit, The Netherlands, [email protected] A. Hordijk, Leiden University, The Netherlands, [email protected]

We consider the expected accumulated costs over a cycle of a phase-homogeneous random walk. For this model we establish sufficient conditions for the existence of the derivative of the cycle cost and we establish an unbiased gradient estimator. The main stability condition for our analysis is that the expected cycle costs

98 TB04 are finite. We thereby improve the results known in the literature so far, where usually finiteness of higher moments of the cycle length is assumed in order to establish unbiasedness of a particular gradient estimator. Our analysis provides minimal conditions for unbiasedness of a gradient estimator for the derivative of the accumulated costs. We will illustrate this with an example where we can establish the existence of the derivative of the expected accumulated costs even though the second moment of the cycle times fails to exist.

Session TB04 - Asymptotic Analysis 2, Chair: Xia in CZ 11

Are end-to-end acknowledgements causing power law delays in large multi-hop net- works?

P.R. Jelenkovic´, Dept. of Electrical Engineering, Columbia University, New York City, USA, [email protected] J. Tan, Dept. of Electrical Engineering, Columbia University, New York City, USA, [email protected]

An end-user that is surfing the Web downloads documents from potentially diverse physical locations, implying that each of the downloads is likely conducted over a new routing path with a different number of hops. Furthermore, the Web downloads are primarily carried over the TCP/IP networks that employ the end-to-end acknowledgements (ACKs), meaning that every time a packet is lost it needs to be resend from the source, which may be highly inefficient on long multi-hop routes that are typically encountered in wireless ad hoc networks. To capture the effects of the variability of routing paths and end-to-end ACKs, we study the delay in a sequence of random number, albeit fixed over each session, of tandem queues with finite buffers and end-to-end ACKs. We show that this model may result in power law delays even if all the parameters of the model are exponentially bounded.

Stochastic ordering for regulated flows and the Better-than-Poisson Property

Fabrice M. Guillemin, France Telecon R & D, Lannion,France, [email protected] Ravi R. Mazumdar, University of Waterloo, Waterloo, Canada, [email protected] Catherine P. Rosenberg, University of Waterloo, Waterloo, Canada, [email protected] Yu Ying, Juniper Networks, Cupertino, Ca, USA, [email protected]

Regulated flows are very useful and common models in the context of traffic control in networks. The most common model is the leaky bucket or (σ, ρ) regulated flows due first analyzed by Cruz. In the deterministic or worst-case setting an extensive set of results based on the (min, plus) algebra, termed Network calculus, has been developed. From a practical standpoint, such an approach leads to very conservative designs and hence there is a need for developing a methodology to analyze the effects on queueing performance (particularly the end-to-end mean delay) that can exploit statistical independence between flows. One such approach is the Negligible Jitter (NJ) conjecture due to Roberts et al. that states that flows that are Better -than-Poisson (BTP) remain BTP within a network after multiplexing. In this talk we present some partial answers to this question. Specifically we show that leaky-bucket reg- ulated flows satisfy the Better-than-Poisson property at the ingress. To do so we construct the appropriate stochastic ordering since the strong ordering does not hold. We then show that when a large number of inde- pendent flows are multiplexed, the regulated flows retain their initial regulation parameters with very high probability within the network. In particular, for concentrator networks, one can show that the end-to-end mean delay can be bounded by using link independence coupled with the BTP property.

99 11 – Abstracts

Sharpness: a sufficient and necessary condition for throughput scalability

A. Chaintreau, Thomson, France, [email protected]

This paper describes a new class of distributed discrete-event systems described as infinite collections of tasks, each requiring a random time, that should follow some precedence rules. Our main assumptions is that the tasks are organized along an euclidean lattice, and such that the precedence rules are invariant by translation. This model includes a broad class of regular communication networks such as tandem of queues, which may implement some local synchronization between servers. Based on percolation results, we prove the following throughput scalability result: The subset of completed tasks grows with time according to a positive speed in any direction. The result is shown to derive from only two conditions: (1) a criterion called “sharpness” satisfied by the precedence relation and (2) a general moment condition on each task completion time, which only depends on the dimension of the lattice. As an application, the results we present offer general guidelines to design scalable reliable distributed applications on overlay networks.

Scalability of Fork/Join Queueing Networks with Blocking

Cathy H. Xia, Zhen Liu, IBM T.J. Watson Research Center, Hawthorne, NY, USA, {cathyx,zhenl}@us.ibm.com Don Towsley, Dept. of Computer Science, University of Massachusetts, Amherst, MA, USA, [email protected] Marc Lelarge, INRIA & Ecole Normale Supérieure, Paris, France, [email protected]

This paper investigates how the throughput of a general fork/join queueing network with blocking behaves as the number of nodes increases to infinity while the processing speed and buffer space of each node stay unchanged. The problem is motivated by applications arising from distributed systems and computer networks. One example is large-scale distributed stream processing systems where TCP is used as the transport protocol for data transfer in between processing components. Other examples include reliable multicast in overlay networks, and reliable data transfer in ad hoc networks. Using an analytical approach, the paper establishes bounds on the asymptotic throughput of such a network. For a subclass of networks which are balanced, we obtain sufficient conditions under which the network stays scalable in the sense that the throughput is lower bounded by a positive constant as the network size increases. Necessary conditions of throughput scalability are derived for general networks. The special class of series-parallel networks is then studied in greater detail, where the asymptotic behavior of the throughput is characterized.

Session TB05 - Randomized Algorithms and Resource Control Schemes, Chair: Borst in CZ 12

Optimal peer-to-peer broadcasting schemes

Laurent Massoulié, Thomson, Paris Research Lab, [email protected] Andy Twigg

This talk focuses on the problem of relaying a live stream of information from a source to nodes of a peer-to-peer system. Scheduling rules based on local information are proposed, and shown to achieve the theoretical broadcasting limit. A classical characterization, due to Edmonds, of the number of spanning trees that can be packed in a graph is retrieved as a corollary. We next discuss a more general scenario where several live streams of information are to be disseminated concurrently. Local rules are again proposed and shown to achieve the theoretical limit. As a byproduct we obtain an extension of Edmonds’ theorem on packing of spanning trees to multiple commodities.

100 TB05

Mean field limits of particle systems in varying environments, Application to mod- eling random multi-access algorithms

C. Bordenave, Department of Electrical Engineering, University of California, Berkeley, US. [email protected] D. McDonald, Department of Mathematics and Statistics, University of Ottawa, Canada. [email protected] A. Proutière, KTH, Radio Communication Systems, Sweden. [email protected]

We study an interacting particle system whose dynamics depends on an interacting random environment. As the number of particles grows large, the transition rate of the particles slows down (perhaps because they share a common resource of fixed capacity). The transition rate of a particle is determined by its state, by the empirical distribution of all the particles and by a rapidly varying environment. The transitions of the environment are determined by the empirical distribution of the particles. We prove the propaga- tion of chaos on the path space of the particles and establish that the limiting trajectory of the empirical measure of the states of the particles satisfies a deterministic differential equation. This deterministic dif- ferential equation involves the time averages of the environment process. We apply our results to analyze the performance of communication networks where users access some resources using random distributed multi-access algorithms. For these networks, we show that the environment process can be approximated by a process describing the number of clients in a certain loss network, which allows us provide simple and explicit expressions of the network performance.

Metastability of CDMA cellular systems

N. Antunes, Universidade do Algarve, Faculdade de Ciências e Tecnologia, Faro, Portugal, [email protected] C. Fricker, INRIA, domaine de Voluceau, 78153 Le Chesnay Cedex, France, [email protected] Ph. Robert, INRIA, domaine de Voluceau, 78153 Le Chesnay Cedex, France, [email protected] D. Tibi, Université Paris 7, UMR 7599, 2 Place Jussieu, 75251 Paris Cedex 05, France, [email protected]

In this it is shown that the coexistence of a variety of different traffics in third generation cellular networks may lead to a very undesirable behavior of the whole network: a metastability property. When this property holds, the state of the network fluctuates on a very long time scale between different set of states. These long oscillations of the network make impossible to predict the average performances of some of the key characteristics of the connections, such as the handoff blocking rate or the probability of call blocking. As a consequence, the quality of service provided by such a network can be guaranteed only by, sometimes poor, lower bounds. Experiments of networks with this behavior are presented and the analysis of a corre- sponding simplified mathematical model is developped. The practical implications in the design of radio resource management for CDMA cellular networks are discussed.

Ranking and suggesting tags in collaborative tagging applications

Milan Vojnovic, Microsoft Research, United Kingdom, [email protected]

We consider collaborative tagging systems where users can attach tags to information objects. Such sys- tems are widely used to add keywords meta-data to photos, videos, or web pages (social bookmarking applications). The meta-data is then used by information retrieval mechanism to provide accurate query answers. To that end, the goal of collaborative tagging systems is to quickly discover the true ranking of a tag for an information object with respect to a given ranking criteria. In this paper, we consider the pop- ularity rank as a ranking criteria. Many collaborative tagging systems help users tagging of an object by making suggestions based on the tagging history of an information object. The problem with making tag

101 11 – Abstracts suggestions is that they may reinforce some tags and the system fails to discover the true popularity rank of a tag for a given information object. To investigate this issue, we propose and study several algorithms for ranking and suggesting tags in collaborative tagging systems where we focus on the following design objectives: (a) learn true popularity rank, (b) make relevant suggestions, and (c) learn fast. We find that simple incremental updates of the suggestion set, which besides suggestion set size require no configuration constants, offer good performance. Performance is evaluated by analysis and numerical results for which we used a dataset of complete tagging histories of urls that we crawled from a popular social bookmarking web service over a month period. Joint work with James Cruise, Dinan Gunawardena, and Peter Marbach.

Session TB06 - Many-server Models 2, Chair: Van Leeuwaarden / Zwart in CZ 13

Corrected diffusion approximations for many-server queues I

A.J.E.M. Janssen, Philips Research, The Netherlands, [email protected] Johan S.H. van Leeuwaarden, Eindhoven University of Technology, The Netherlands, [email protected] Bert Zwart, Georgia Institute of Technology, USA, [email protected]

Queues with many servers are relevant in the performance analysis of large call centers. Since the com- plexity of such queueing models grows as the number of servers grows large, it is desirable to have scalable performance analysis tools available. A popular tool that offers such scalability is the Halfin-Whitt approx- imation of the no-wait probability in the M/M/s queue. The goal of this work is to improve the Halfin-Whitt approximation of the delay probability such that the resulting formula remains accurate when the number of servers is not necessarily large, and/or when the system is not necessarily in heavy-traffic. In addition, we give analytic estimates of the accuracy of the celebrated square root staffing rule. This talk will be the first in a series of two. In the second talk, we present similar results for the M/D/s queue.

Corrected diffusion approximations for many-server queues II

A.J.E.M. Janssen, Philips Research, The Netherlands, [email protected] Johan S.H. van Leeuwaarden, Eindhoven University of Technology, The Netherlands, [email protected] Bert Zwart, Georgia Institute of Technology, USA, [email protected]

To obtain insight in the quality of heavy-traffic approximations for queues with many servers, we consider the steady-state number of waiting customers in an M/D/s queue as s → ∞. In the Halfin-Whitt regime, it is well known that this random variable converges to the supremum of a Gaussian random walk. We develop methods that yield accurate results in terms of series expansions and inequalities for the probability of an empty queue, and the mean and variance of the queue length distribution. The main idea is to view the M/D/s queue through the prism of the Gaussian random walk: as for the standard Gaussian random walk, we provide a scalable series expansion for the emptiness probability involving terms that include the Riemann zeta function. This talk will be the second in a series of two. In the first talk, we present similar results for the M/M/s queue.

102 TB07

A new view on the G/GI/N queue in Halfin-Whitt regime

J.E Reed, New York University, United States, [email protected]

We study the G/GI/N queue from the perspective of the servers in the system. This point-of-view allows us to obtain results for the G/GI/N queue in the Halfin-Whitt regime which were previously proven through different means. Our main results are to obtain weak limits for both the fluid and diffusion scaled queue length process.

Moment Equivalences Between Multiserver and Single Server Queues

A. Scheller-Wolf, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA, [email protected] R. Vesilo, Macquarie University, New South Wales, Australia, [email protected]

Generalizing techniques of Foss and Korshunov, we relate the delay-moment behavior of FIFO multiserver queues to the delay-moment behavior of sets of independent FIFO single server queues. This provides (i) alternate derivations of known sufficient conditions for stationary multiserver delay moments; (ii) more general necessary conditions, and; (iii) a structural interpretation underlying these moment conditions. As such we provide intuition about why multi-server queues behave as they do.

Session TB07 - Stochastic Control and PDE, Chair: Atar in CZ 14

Optimal Stopping and Free Boundary Characterizations for some Brownian Con- trol Problems

Amarjit Budhiraja, Department of Statistics, University of North Carolina, [email protected] Kevin J. Ross, Department of Statistics, Stanford University, [email protected]

We consider a two dimensional singular control problem with state constraints. By studying a related optimal stopping problem we establish that the value function is C1. We then characterize an optimally controlled process as a diffusion reflected at an appropriate free boundary. This resolves a conjecture of Martins-Shreve-Soner(1996) on the form of an optimal control for this class of control problems.

A control-theoretic analysis of fast-sweeping methods for static Hamilton-Jacobi equations

Paul Dupuis, Brown University, USA, [email protected]

The term “fast sweeping” refers to a set of numerical methods for static Hamilton-Jacobi equations, with the property that the iterative scheme that is used to solve for the numerical approximation converges after essentially a fixed number of sweeps through the numerical grid. In this talk we will discuss qualitative properties of these methods that can be proved using a control representation for the iterative scheme.

HJB equations, no arbitrage, and generalized Brownian networks

R. Atar, Technion, Israel, [email protected] A. Budhiraja, University of North Carolina at Chapel Hill, NC, [email protected] R.J. Williams, University of California at San Diego, California, [email protected]

103 11 – Abstracts

Let X(t) = X(0)+ B(t)+GU(t), t ≥ 0, where B is an n-dimensional Brownian motion, G a fixed matrix, and U a ‘singular control’, i.e., an adapted process that is locally of bounded variation and keeps X in a given domain in Rn at all times t ≥ 0. Under some assumptions, a Hamilton-Jacobi-Bellman equation characterizes the value of a control problem in which a cost is to be minimized over all such controls U. The existence and uniqueness of solutions to this equation are shown to be connected to the intuitive ‘no arbitrage’ condition, that states that an instantaneous control action that does not affect the process X does not result in any immediate reduction in cost. Our results apply, in particular, to the diffusion control problems derived by Harrison and Williams (2005) for Generalized Brownian networks.

A Representation for the Characteristic Functional of Banach Valued Stable Ran- dom Measures

S. Mahmoodi, Isfahan University of Technology, Iran, [email protected] A.R. Soltani, Kuwait University, State of Kuwait, [email protected]

A Banach space (X: real separable) valued symmetric α stable random measure 8 with independent increments on a measurable space (F, F ) is considered. A pair (k, µ), called here a control pair, for + which k : X × F → R , µ a positive measure on (F, F ), is introduced. It is proved that the law of 8 is governed by a control pair; and every control pair will induce such 8. Moreover k is unique for a given µ. Our derivations are based on the Generalized Bochner Theorem and the Radon- Nikodym Theorem for vector measures.

Session TB08 - Fluid Models, Chair: Scheinhardt in CZ 15

Analysis of a resource sharing policy for computer systems using fluid queues

N. Gautam, Texas A&M University, USA, [email protected] S.R. Mahabhashyam, Oracle Corporation, USA, [email protected]

A system consisting of two buffers, each with independent fluid sources is considered in this research. The output capacity for the two buffers depend on the workload of only one of the buffers which is observable. A threshold-based policy is considered to dynamically assign output capacities for both buffers. Marginal workload distributions for the two buffers need to be evaluated for this policy. The key contribution of this work is the performance analysis to derive the workload distribution in the two buffers. For the first (observable) buffer we use a spectral analysis by writing out the partial differential equations for the buffer content. For the second (unobservable) buffer we model the random environment using a semi-Markov process and use a compensating source for performance analysis. In addition, the work also provides some guidelines to choose the output capacities for the two buffers as well as a mathematical program to deter- mine an optimal threshold to dynamically switch between output capacities. Further, various applications of such systems to computer-communication networks are discussed.

A feedback fluid queue with two congestion control thresholds

R. Malhotra, Bell Labs Europe, Alcatel-Lucent, Capitool 5, 7521 PL Enschede, The Netherlands, [email protected] M.R.H. Mandjes, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands, [email protected] W.R.W. Scheinhardt, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, [email protected] J.L. van den Berg, TNO and University of Twente, P. O. Box 5050, 2600 GB Delft, The Netherlands, and [email protected]

104 TB08

We present and analyze a fluid queue with a feedback-based traffic rate adaptation scheme which uses two thresholds. Previous work on feedback fluid queues has modeled queues with a single threshold to signal the onset and end of congestion to the rate adapting traffic source. A major drawback of such systems is that they impose significant bandwidth overhead due to signals sent to the source every time the threshold is crossed. In addition, the single threshold makes it difficult to find a good trade-off between throughput and delay performance of the system. In our model with two thresholds, the higher threshold B1 is used to signal the beginning of congestion and can be used to maximize throughput performance. The lower threshold B2 is used to signal the end of congestion and can be used to minimize delay. The difference between the two thresholds helps to control the amount of feedback signals sent to the traffic source. In our model a fluid source is active which can behave like either of two Markov fluid processes. The first applies as long as the upper threshold B1 has not been hit from below. As soon as that happens, the traffic source adapts and switches to the second process, until B2 is hit from above. We present the exact analysis of the stationary distribution of the buffer occupancy and performance measures. Important novelty of our model compared to earlier feedback models is that the traffic source may behave in two different ways between the two thresholds, depending on the past of the buffer process.

On shot noise fluid queues and infinite-server queues with batch arrivals: time- dependent behavior

W.R.W. Scheinhardt, University of Twente, The Netherlands, [email protected] R.J. Boucherie, University of Twente, The Netherlands, [email protected] W.F. de Graaf, The Netherlands, [email protected]

We consider a shot noise model, in which a fluid reservoir receives quantities of fluid (with some general probability distribution) that arrive according to a Poisson process. Moreover, the reservoir is continu- ously drained at a rate that is proportional to the current content of the reservoir. We are interested in the time-dependent behavior of the content of the reservoir, where we also allow the arrival rate of the Pois- son process to depend on time. We derive the Kolmogorov forward (partial differential) equation for the Laplace-Stieltjes transform of the content at time t, and solve it. Interestingly, the model may be seen as the limiting case of a certain sequence of infinite-server queues with batch arrivals. We show why this fact holds, and how it enables us to find the solution also from the (known) generating function of the number of jobs in the infinite-server queue. Some examples show how the moments of the content process can behave as a function of time.

Tandem Queue with Server Slow-down

D.I. Miretskiy, University of Twente, The Netherlands, [email protected] W.R.W. Scheinhardt, University of Twente, The Netherlands, [email protected] M.R.H. Mandjes, University of Amsterdam, The Netherlands, [email protected]

We study how rare events happen in the standard two-node tandem Jackson queue and in a generalization, the socalled slow-down network. In the latter model the service rate of the first server depends on the number of jobs in the second buffer: the first server slows down if the amount of jobs in the second buffer is above some threshold and returns to its normal speed when the number of jobs in the second buffer is below the threshold. This property protects the second buffer from frequent overflow. We provide the (non-trivial) stability condition for this model. Considering the joint queue length process for both models we construct the most probable path to some rarely visited subset of the state space, starting from any initial position, when the system is scaled with some large parameter. Based on this knowledge we con- struct state-dependent Importance Sampling schemes, which provide fast and precise results for estimating probabilities of rare events of interest (for example, overflows of a single node).

105 11 – Abstracts

Session TB09 - Supply Chains, Chair: Ettl / De Kok in CZ 16

Centralized and decentralized vendor-managed inventory optimization under dif- ferent levels of information sharing

M. Ettl, IBM Thomas J. Watson Research Center, USA, [email protected] Y. Lu, IBM Thomas J. Watson Research Center, USA, [email protected] M.S. Squillante, IBM Thomas J. Watson Research Center, USA, [email protected]

In this paper we study the trade-offs between serviceability, liability, and inventory in an outsourced supply chain with vendor-managed inventory (VMI). We formulate different stochastic optimization problems to characterize the interactions between different supply chain echelons through their serviceability and liability arrangements under different types of control and information sharing assumptions. Structural and algorithmic analysis of the optimization problems enables us to reveal the connections between the interplay of serviceability and liability and the control of inventory at each echelon. We also present extensive numerical experiments to illustrate and quantify our theoretical findings.

Optimal Policies for Joint Replenishment and Substitution

H. Xu, School of Management, Huazhong University of Science and Technology, Wuhan, China D.D. Yao, IEOR Department, Columbia University, New York, NY 10027, USA, [email protected] S. Zheng, Department of Information and Systems Management, Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong,[email protected]

We study an inventory model that allows substitution among products. Replenishment orders of products must be made at the beginning of the selling season. The products are then used to supply demand for the entire season of N periods. When a product is out of stock, the retailer may offer a substitution, which may or may not be accepted by the customer. The decisions are the replenishment quantity for each product at the beginning of the season, and the dynamic substitution rule in each period of the season. We first establish the optimality of a threshold type substitution rule for a two-product model, and show that the objective function is both concave and supermodular in the order quantities. We also consider constraints on the order quantities, and solve a related yield management problem. These results also form the basis for a heuristic policy for the case of multiple products.

Benefits and risks of price protection in the computer industry

M. Ettl, IBM Thomas J. Watson Research Center, USA, [email protected] P. Huang, IBM Thomas J. Watson Research Center, USA, [email protected] R. Kapuscinski, Stephen M. Ross School of Business, University of Michigan, Ann Arbor, MI 48109, USA, [email protected] K. Sourirajan, IBM Thomas J. Watson Research Center, USA, [email protected]

In high-tech industries where technological obsolescence is high, price protection has become a standard element of contracts between manufacturers and distributors. The main idea of price protection is to credit distributors for price erosion of goods that were purchased up to l weeks before a price drop but not yet sold to end customers. At the time price change is announced, the manufacturer reviews the distributor’s purchases over the last l weeks and at the distributor’s current inventory. The distributor is given a price protection credit equal to the product of the smaller of the two quantities and the price decrease. While price protection was intended to give distributors sufficient incentives to stock enough inventory, empirical evidence suggests that price protection often leads to overstocking. We analyze acceptable mechanisms to reduce excess inventory caused by price protection.

106 TC01

Period Order Quantities in Multi-Item Multi-Echelon Inventory Systems

Ton de Kok, Technische Universiteit Eindhoven, The Netherlands, [email protected]

We consider multi-item multi-echelon inventory systems under periodic review echelon order-up-to poli- cies. The review periods of different items may differ. We assume that the review period of an item is a multiple of the review period of each of its parent items. When controlling multi-item multi-echelon sys- tems a key issue is the interaction between parent and child items in case of insufficient stock of child items to satisfy orders from their parents. In case of differences in review periods of parents of a particular child item, this interaction is further complicated by the fact that some parent items may order at some point in time, while other parent items do not. This may lead to unwanted implicit prioritization of items that order "early" in the replenishment cycle of the child item. We propose a child item allocation mechanism that takes differences in review periods of parent items into account, while allowing for rigorous mathematical analysis of the multi-item multi-echelon system.

Tuesday 2:45pm - 4:15pm

Session TC01 - Statistics 2, Chair: Kuhnt in CZ 4

Statistical Analysis of Spatial Patterns of Customer Behaviors

X.X. Bai, IBM China Research Laboratory, [email protected], ZhongGuanCun Software Park, 100094, China W.J. Yin, IBM China Research Laboratory, [email protected], ZhongGuanCun Software Park, 100094, China J. Dong, IBM China Research Laboratory, [email protected], ZhongGuanCun Software Park, 100094, China

Deep insights into customer behaviors is critical in marketing activities, e.g. targeting customers for pro- motion events, recognizing risk customers when competitor entries, etc. However, conventional customer behavior analysis does not take into account of the geographic location and distribution of customers, e.g. typical customer behavior definitions by RFQ (Recency, Frequency and Quantity) can not indicate the fact that people in certain neighborhoods are more likely to shop in a similar way. This work studies the spatial pattern of customer behaviors and can help answer questions, i.e. what kind of customers prefer to buy what merchandise at which format of stores how far to his residential areas. These spatial behaviors pat- terns are examined in a real china retailer case based on statistics on its member card info and transaction data.

On the performance of estimators of parameters in autoregressive model under asymmetric loss

Soma Roychowdhury,Indian Institute of Social Welfare and Business Management and Visva-Bharati University Debasis Bhattacharya, Indian Institute of Social Welfare and Business Management and Visva-Bharati University, [email protected]

In the AR(1) set-up it is well known that for different ranges of the parameters the distributions of different quantities of interest (for example,the estimators,test statistics etc.)are different.There the equal amount of over estimation and under estimation may not have the equal consequences.Thus the use of symmetric loss function in estimating the unknown parameters is not recommendable.Here we have shown that under

107 11 – Abstracts asymmetric loss function Bayes estimator outperforms the classical point estimators when the operational prior is sufficiently diffuse.Prediction problem will also be discussed using asymmetric loss funtion.

On nonparametric inference for a class of queuing systems

M.Pawlak, University of Manitoba, Canada, [email protected]

Nonparametric methods for identification of the G/G/1 - type queuing system are studied. One of many possible scenarios can be described as follows: given observations from interarrival and service time distri- butions of the G/G/1 queuing system we wish to estimate a stationary waiting time distribution function. Yet another case of interest is the form of service time density function given observations of the durations of busy periods. In all these situations one has no any a priori knowledge about the underlying distributions and nonparametric techniques for estimating the density and distribution functions are particularly suited. Unlike standard nonparametric estimation problems the queuing theory is characterized by the fact that we wish to infer about a distribution using indirect (incomplete) information. This can be viewed an inverse problem and in this paper we examine a number of such problems resulting in the context of the G/G/1 queuing system. Nonparametric estimation techniques for the studied inverse queuing problems are pro- posed and their asymptotic properties are established. In particular, expansions of the bias and variance of the estimators are given and asymptotic normality is established. The accuracy of the estimation techniques critically depends on the degree of the invertibility of a given inverse problem.

An application of algebraic statistics to Bowker’s test for symmetry

S. Kuhnt, Eindhoven University of Technology, The Netherlands, [email protected] A. Krampe, University of Dortmund, Germany, [email protected]

Bowker (1948) proposed a commonly used test for axial symmetry in contingency tables. However, the used chi-squared approximation of the distribution of the test statistic might not always be adequate, for example if the table is sparse. Therefore, modifications of the test statistic have been discussed. We propose a test of symmetry using results from algebraic statistics. Diaconis and Sturmfels (1998) combine computational commutative algebra and statistics to simulate samples of the conditional distribution of a discrete exponential family with given sufficient statistic. In particular Groebner bases are utilized for the construction of the Markov chain. We carry out a simulation study to determine and compare the performances of the simulation test, the Bowker test and two modifications. References Bowker, A.H., 1948. A Test for Symmetry in Contingency Tables. Journal American Statistical Association 43, 572–574. Diaconis, P.; Sturmfels, B., 1998. Algebraic Algorithms for Sampling from Conditional Distributions. Annals of Statistics 26, 363– 397. Krampe, A.; Kuhnt, S., 2007. Bowker’s Test for Symmetry and Modifications within the Algebraic Framework. To appear in Computational Statistics & Data Analysis.

Session TC02 - Anomalous Diffusion, Chair: Metzler in CZ 5

Anomalous Subdiffusion Revisited

Joseph Klafter, Tel Aviv University, [email protected]

An overview is given on anomalous diffusion processes with emphasis on subdiffusion, which is widespread in many complex systems. We consider in particular the case of subdiffusion which corresponds to a continuous-time random walk process in which the waiting time for a step is given by a probability distri- bution with a diverging mean value. Relationships to the fractional diffusion or Fokker-Planck equations

108 TC03 and their generalizations will be discussed. Special attention will be given to the problem of the initial conditions and the corresponding aging processes. We will highlight recent results on linear response of a continuous-time random walk system to a time-dependent field.

Anomalous Diffusion Leads to Weak Ergodicity Breaking

E. Barkai, Bar-Ilan University Ramat Gan 52900 Israel, [email protected]

Time averages of physical observables of several stochastic processes describing anomalous diffusion, remain random even in the limit of long measurement time. Examples include blinking quantum dots, anomalous diffusion of mRNA in the cell, and the sub-diffusive continuous time random walk model (CTRW). The basic question for such systems is what statistical theory replaces standard ergodic statistical mechanics? A theory of weak ergodicity, based on the multi-dimensional arcsine distribution, gives the distribution of the time averages of physical observables for such systems. When the diffusion turns normal, usual ergodic behavior is found.

Mesoscopic approach to reactions between subdiffusing particles

Igor M. Sokolov, Humboldt University, Berlin, [email protected]

We consider a system of subdiffusive particles which react locally according to the mass action law. We show that the correct system of reaction-subdiffusion equations for this case is not obtained by adding a reaction term to the fractional diffusion equations describing subdiffusion and discuss the particular form of reaction-subdiffusion equations for the case of irreversible reactions. We first consider the A→B reaction as the simplest example to check the formalism, since in this case the result can be obtained using an alternative approach. We then turn to a more complex situation of a stationary A+B→0 reaction under constant concentrations of reactants on both sides of the system in a slab geometry. We show that the structure of reactants’ distribution in the case of subdiffusion differs vastly from the one in the normal diffusion case. The most important differences correspond to the formation of complex reaction profiles with accumulation and depletion zones and to their extreme sensitivity to the boundary conditions.

Fractional Fokker-Planck dynamics: Stochastic representation and computer sim- ulation

Aleksander Weron, Wroclaw University of Technology, [email protected] M. Magdziarz

A computer algorithm for the visualization of sample paths of anomalous diffusion process is developed. It is based on the stochastic representation of the fractional Fokker-Planck equation describing anoma- lous diffusion in a nonconstant potential. We also answer positively a question raised by R. Metzler and J.Klafter (2000): Can one see a competition between subdiffusion and Levy flights in the framework of the fractional Fokker-Planck dynamics ? Our method of Monte Carlo simulations demonstrates the competi- tion on the level of realizations as well as on the level of probability density functions of the anomalous diffusion process. The introduced algorithm will surely provide tools for studying many relevant statistical characteristics of the fractional Fokker-Planck dynamics.

Session TC03 - Cross-entropy, Chair: Kroese in CZ 10

Parametric Minimum Cross-Entropy Method

R. Y. Rubinstein, Technion, Israel Institute of Technology, Israel, [email protected]

109 11 – Abstracts

A. Dolgin, Technion, Israel Institute of Technology, Israel, [email protected] D. P. Kroese, The University of Queensland, Australia, [email protected] P. W. Glynn, Stanford University, USA, [email protected]

We introduce a new method, called the parametric minimum cross-entropy (PME) method, for rare event probability estimation and counting with a particular emphasis on the satisfiability problem. The PME method is derived from the well known Kullback’s minimum cross-entropy method, called MinxEnt. It is based on the marginal distributions derived from the optimal joint MinxEnt distribution and is a parametric version of it. Similar to cross-entropy (CE) method, the PME algorithm first casts the underlying counting problem into an associated rare-event probability estimation problem, and then finds the optimal parameters of the importance sampling distribution to estimate efficiently the desired quantity. We present supportive numerical results for counting the number of satisfiability assignments and compare PME with the CE method. Our numerical results suggest that the PME method is superior to CE. This is based on the fact that for the satisfiability problem and some other ones involving separable functions the optimal parameters of the importance sampling distribution can be estimated better with the MinxEnt type procedure, rather than with indicators.

A Nonparametric Extension of the Cross-Entropy method

Z.I. Botev∗, The University of Queensland, Australia, [email protected] T. Taimre, The University of Queensland, Australia, [email protected] D.P. Kroese, The University of Queensland, Australia, [email protected]

The Cross-Entropy method has been used extensively as an algorithm for multi-extremal optimization. One key characteristic of the method is the use of a parametric sampling density at each iteration, the pa- rameters of which are updated optimally in some well-defined sense. In this talk we propose the use of a non-parametric kernel density estimator instead of a parametric sampling density. It is well known that the performance of a non-parametric estimator depends crucially on the appropriate choice of the so called bandwidth parameters. One challenge then with the non-parametric model is to optimally select these bandwidth parameters in the context of the Cross-Entropy method. We explain the theory behind the op- timal updating of the bandwidth parameters and provide numerical results demonstrating the effectiveness of the method. The numerical examples include rare-event estimation and continuous multi-dimensional optimization.

A Generalized Cross-Entropy Method: Discrete State Spaces

T. Taimre, The University of Queensland, Australia, [email protected] Z. I. Botev, The University of Queensland, Australia, [email protected] D. P. Kroese, The University of Queensland, Australia, [email protected]

Combinatorial optimization and difficult counting problems are encountered in many diverse fields. Recent work on the cross-entropy method, the minimum cross-entropy method, and its parametric modification, has led to the development of easy-to-implement randomized algorithms for solving such problems. One major feature of these existing techniques is the use of parametric sampling distributions. These lead to analytic updating rules, but are often unable to adequately approximate the desired target densities. In this talk, we extend the existing ideas of the cross-entropy method to formulate a generalized cross-entropy methodology, built around the idea of using non-parametric sampling distributions that are, in some sense, optimal. We illustrate this extension for optimization and estimation problems on discrete state spaces, with examples including algorithms for solving MAX-CUT and for estimating the number of solutions of MAX-CUT.

110 TC04

Adaptive importance sampling algorithms for rare-event simulations of the infinite- server queue.

A. Ridder, Vrije Universiteit Amsterdam, The Netherlands, [email protected]

In this paper we consider importance sampling simulations for the estimation of the transient probability that the infinite-server queue with Poisson arrivals and general service-time distributions reaches high levels within a predefined time window. Under the light traffic assumption this is a rare event. Moreover, we assume that the service times are highly variable. In case of exponential servers we have showed that the optimal path approach leads to a time-dependent, exponentially tilted, asymptotically optimal change of measure. For general service times we have developed an adapted version of this change of measure which gives good performances of the importance sampling estimators in case of low-variable service times. For high-variable service times we consider a further adjustment of the algorithm by applying cross-entropy techniques.

Session TC04 - Limit Theorems for Queueing Systems with Measure- valued State Descriptors, Chair: Puha in CZ 11

Limit Theorems for Many-Server Queues

K. Ramanan, Carnegie Mellon University, Pittsburgh, [email protected]

Many-server queues arise in a variety of applications, including computer data networks and call centers. We uniquely characterise the strong law of large numbers or fluid limit (as the number of servers goes to infinity), of several functionals of this system, including in particular the total workload process. We also identify the large time behavior of the asymptotic limit process. The proof involves a measure-valued description of the Markovian dynamics and the analysis of a certain (distributional) partial differential equation with boundary conditions. This is joint work with Haya Kaspi.

Heavy Traffic Analysis for EDF Queues with Reneging

Lukasz Kruk, Maria Curie-Sklodowska University, Lublin Poland, [email protected] John Lehoczky, Carnegie Mellon University, Pittsburgh, PA USA, [email protected] Kavita Ramanan, Carnegie Mellon University, Pittsburgh, PA USA, [email protected] Steven Shreve,Carnegie Mellon University, Pittsburgh, PA USA, [email protected]

We present a heavy-traffic analysis of the behavior of a single server queueing system in which customers have deadlines, are served in EDF order, and renege at the moment their deadline elapses, even if service is in progress. The performance of the system is measured by the fraction of customer missing their deadline. This extends prior work in which customers were served to completion, even if they were late. With reneging the scaled workload process converges to a doubly-reflected Brownian motion process, and the scaled measure-valued workload process converges to a nonrandom function, which is characterized. Simulation results are presented to compare the performance of the systems with and without reneging.

Central Limit Theorems for a large queuing network with resource pooling

C. Graham, CNRS and École Polytechnique, France, [email protected]

Consider a network in which customers choose uniformly L ≥ 2 queues among N and join the shortest. A natural state descriptor is the tail of the empirical measure of the queue lengths, for which Vvedenskaya,

111 11 – Abstracts

Dobrushin and Karpelevitch proved a law of large numbers (LLN) result in equilibrium as N goes to infinity. The limit tail is hyper-exponentially decreasing, a fact of significant applied interest for such a simple resource pooling scheme. We provide convergence rates and confidence intervals through a corresponding central limit theorem (CLT). The invariant laws are known only implicitly, and we first prove a functional CLT for transitory regimes in which the initial conditions satisfy a CLT. We then study the long time behavior for the limit Ornstein-Uhlenbeck process. The functional CLT in equilibrium is eventually derived by proving rigor- ously that we may interchange the long time and the large network limits. An essential tool is a global exponential stability result for the nonlinear dynamical system governing the LLN limit. It is based on original comparison techniques with linear systems, combined with the study of the spectral decomposi- tion and spectral gap of the latter, performed in well-chosen Hilbert spaces.

The Fluid Limit of a Shortest Remaining Processing Time Queue

D.G. Down, McMaster University, Department of Computing and Software, 1280 Main Street West, Hamilton, ON Canada L8S 4K1, [email protected] H.C. Gromoll, Department of Mathematics, University of Virginia, Charlottesville, VA 22903, [email protected] A.L. Puha, Department of Mathematics, California State University San Marcos, San Marcos, CA 92096-0001, [email protected]

Consider a GI/GI/1 queue operating under shortest remaining processing time with preemption. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. Of particular interest is the waiting time for large jobs, which can be tracked using the frontier process, the largest service time of any job that has ever been in service. We propose a fluid model and present a functional limit theorem justifying it as an approximation of this system. The fluid model state descriptor is a measure valued function for which the left edge of the support is the fluid analog for the frontier process. Under mild assumptions, we prove existence and uniqueness of fluid model solutions. Furthermore, we are able to characterize the left edge of fluid model solutions as the right continuous inverse of a simple functional of the initial condition, arrival rate, and service time distribution. When applied to various examples, this characterization reveals the dependence on service time distribution of the rate at which the left edge of the fluid model increases.

Session TC05 - Communication Systems 2, Chair: Choi in CZ 12

Performance analysis of small non-uniform packet switches

P. Beekhuizen, Philips Research / Eurandom, The Netherlands, [email protected] J.A.C. Resing, Eindhoven University of Technology, The Netherlands, [email protected]

Due to the increasing number of modules on chips, e.g., memory blocks, processors, etc., traditional buses between such modules no longer suffice. As an alternative paradigm, so-called Networks on Chips (NoCs) are emerging, where modules are connected using switches or routers. These networks resemble telecom- munication networks such as LANs or ATM networks. One major difference, however, with LANs and ATM networks is the fact that the number of input ports in a NoC switch is usually quite small (say 4 or 5). This especially invalidates the asymptotic analyses commonly performed. In our talk we analyse the performance of a small non-uniform packet switch. First, we analyse the situation where all queues are saturated using a Markov chain approach. This analysis gives rise to a throughput approximation for the non-saturated case, as well as an approximation of the stability conditions per queue. Second, we approximate the service rate in moderate traffic by a quadratic interpolation between its value

112 TC05 in light traffic and in saturation. The service rate found is then used as a parameter of a mean waiting time approximation.

Performance analysis of sleep mode and idle mode for best effort traffic in IEEE 802.16e

Yong Hyun Lee, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] Eunju Hwang, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] Kyung Jae Kim, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] Jung Je Son, Global Standards & Research Team, Telecom R&D Center, Samsung Electronics, Suwon, Korea, [email protected] Bong Dae Choi, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected]

The IEEE 802.16e is designed to use sleep mode and idle mode for power saving of a mobile station. We apply the sleep mode operation of power saving class of type 1 when MS is in on-session and the idle mode operation when MS is in off-session. We obtain the performance measures such as power consumption ratio and mean delay of packets. We show that analytic results match with simulation results quite well. Numerical examples show that the sleep mode and idle mode provide a considerable reduction on energy of mobile station.

Performance Analysis of Power Saving Class of Type 1 for both Downlink and Up- link Traffics in IEEE 802.16e

Sangkyu Baek, Korea University, Korea, [email protected] Jung Je Son, Samsung Electronics, Korea, [email protected] Seon Mi Lee, Korea University, Korea, [email protected] Bong Dae Choi, Korea University, Korea, [email protected]

We consider the power saving class of type 1 in IEEE 802.16e where a mobile station communicates with a base station by both downlink and uplink traffics. This system is modelled as an embedded Markov chain and a semi-Markov process. We obtain power consumption ratio and queueing delay. We show that our analytic results match with simulation results very well. Using our performance analysis, we find the optimal system parameters such as the initial sleep window, the final sleep window and the close-down time which minimize the power consumption of the mobile station while satisfying the required QoS on mean delay.

A New Power Saving Mechanism with Periodic Traffic Indications in the IEEE 802.16e

Eunju Hwang, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] Kyung Jae Kim, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected] Jung Je Son, Global Standards & Research Team, Telecom R&D Center, Samsung Electronics, Suwon, Korea, [email protected] Bong Dae Choi, Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Korea, [email protected]

113 11 – Abstracts

A new sleep mode scheme called the power saving mechanism with periodic traffic indications in the IEEE 802.16e. In the proposed scheme, traffic indication(TRF-IND) messages are regularly sent at every constant time to initiate transmission, and the bandwidth and energy can be saved by not sending sleep request(MOB-SLP-REQ) and sleep response(MOB-SLP-RSP) messages which are required in the original power saving classes in the IEEE 802.16e standard. We obtain sleep mode ratio and power consumption ratio by computing the Laplace Stieltjes transforms(LSTs) of the length of awake interval and sleep interval, and total delay by computing the LST of queueing delay. We show that our analytic results match with simulation results very well. Using our performance analysis we find the optimal system parameters such as a TRF-IND interval and a close-down time which minimize the power consumption of MS while satisfying the required quality of service(QoS) on delay. Finally we show that the proposed scheme performs better then the power saving class of type 1 in the IEEE 802.16e standard.

Session TC06 - Matrix-Analytic Methods, Chair: Squillante / Taylor in CZ 13

Bounds and Decay Rate for a QBD Process with Countably Many Background States

H. Li, Mount Saint Vincent University, Canada, [email protected] Q.-M. He, Dalhousie University, Canada, [email protected] Y.Q. Zhao, Carleton University, Canada, [email protected]

In this talk, we introduce upper and lower bounds for the tail in the stationary joint distribution and a marginal distribution for a quasi-birth-and-death process with countably many background (phase) states. We also discuss conditions for exactly geometric-tailed and light-tailed, but not exactly geometric, distri- butions for this type of system. Using examples, we demonstrate how to apply these conditions to models of interest.

Algorithmic approach to the extinction probability of branching processes

S. Hautphenne, Université Libre de Bruxelles, Belgium, [email protected] G. Latouche, Université Libre de Bruxelles, Belgium, [email protected] M.-A. Remiche, Université Libre de Bruxelles, Belgium, [email protected]

A large number of questions in fields like biology and telecommunications may be modeled as continuous- time multi-type branching processes. In order to determine the extinction probability for this kind of processes, it is necessary to find the minimal nonnegative solution of a non-linear matrix fixed point equa- tion. We examine the question of solving that equation, using a mixture of algorithmic and probabilistic arguments. This allows us to analyze the efficiency of various algorithms on the basis of the behavior of the branching process itself. We focus our attention here on a particular class of continuous-time multi-type branching processes, named Markovian binary trees (MBT), but our approach is amenable to more complex systems. In the special case of the MBT, the fixed point equation turns out to be quadratic, which makes its resolution particularly clear. Specifically, we analyze three linear algorithms, of which one is new, and, we propose a quadratic algorithm arising from Newton’s iteration method for fixed-point equations. We also give them a probabilistic interpretation in terms of the MBT.

Physical interpretations of algorithms for stochastic fluid flows

Nigel Bean, University of Adelaide, Australia, [email protected] Małgorzata O’Reilly, University of Tasmania, Australia, [email protected] Peter Taylor, University of Melbourne, Australia, [email protected]

114 TC07

We discuss several algorithms, including quadratically convergent algorithms, which can be used to cal- culate the Laplace-Stieltjes transforms of the time taken to return to the initial level in the stochastic fluid flow model. We give physical interpretations of the algorithms within the fluid flow model. These physical interpretations are built upon the physical interpretations of the generator of the Laplace-Stieltjes trans- form of the total amount of fluid that flowed in and out the buffer. The major benefit of understanding these physical interpretations is the ablity to analyze the model using direct, elegant methods, within the fluid flows.

Invariant measures for transient fluid queues

Sarah Van Lierde, Université Libre de Bruxelles Ana da Silva Soares, Université Libre de Bruxelles Guy Latouche, Université Libre de Bruxelles, Belgium

Positive recurrent fluid queues have a matrix-exponential stationary distribution. We show that a transient fluid queue also has a matrix-exponential invariant measure which is not summable. We give a construc- tion for this measure and we give the probabilistic interpretation for the key matrix which appears in the construction.

Session TC07 - MDP 2, Chair: Örmeci in CZ 14

A Semi Markov Decision Approach for Train Conflict Resolution

Assil Al-Ibrahim, University of Amsterdam, The Netherlands, [email protected] Jan van der Wal, Eindhoven University of Technology, The Netherlands, [email protected]

Railways are a common way of transportation. The passengers that use railways demand an increasing level of service. Punctuality plays here a great role. This paper addresses the delays that are a consequence of a conflict between two or more trains. Trains from more than one direction come together and from that point onwards share what we will call the ’destination track’. When two or more trains try to enter the ’destination track’, a decision is to be made about which train to move first. The optimal decision takes into account several factors. We distinguish the train-speeds, the acceleration rate giving their mass, the urgency of trains, previous decisions and the trains that are still to arrive at the junction. To solve this problem we use a Semi-Markov Decision (SMD) approach. An advantage of the SMD-model is that it is able to deal with the dynamics of the train system. A disadvantage is that a description as a SMD, tends to result in systems with tremendously large state spaces. As will be shown, it is possible to find a balance between these two aspects. By means of simulation, the performance of the SMD-model is compared to that of FCFS and other rules.

Exact finite approximations of average-cost countable Markov Decision Processes

Arie Leizarowitz, Faculty of Mathematics, Technion, Haifa, Israel, [email protected] Adam Shwartz, Faculty of Electrical Engineering, Technion, Haifa, Israel, [email protected]

For a countable-state, compact action space Markov decision process we introduce a method to derive em- beddings which produce finite-state Markov decision processes. Each finite-state embedded process has the same optimal cost as the original model—and similarly, costs of any stationary Markov policy agree. Moreover, the embedded Markov decision process agrees with the original process when restricting to the approximating finite set, in the sense that the available actions are the same, and the use of corresponding actions yields identical transition probabilities and identical immediate rewards.

115 11 – Abstracts

The embedded process can be used as an approximation which, being finite, is more convenient for com- putation and implementation. Since the optimal costs are the same, it is an approximation only in the sense that the process is different outside the finite set. The size of the approximation determines the range of states on which the approximation has the same dynamics as the original process. This method extends easily to constrained MDPs. The ideas also apply to optimal exit problems.

A Controlled Markov Process Model for a Benchmark Reentrant Manufacturing Line with Infinite and Finite Capacity Buffers

José A. Ramírez-Hernández, University of Cincinnati, USA, [email protected] Emmanuel Fernandez, University of Cincinnati, USA, [email protected]

We present results on the optimal job sequencing control policy for a benchmark reentrant manufactur- ing line, for Markov Decision Models with both infinite and finite capacity buffers, and under an infinite horizon discounted cost criterion. For a set of one-stage cost functions the results obtained provide both sufficient conditions and structural properties for the optimal policy.

Effects of System Parameters on the Optimal Policy Structure in a Class of Queue- ing Control Problems

E.L. Örmeci, Koç University, Turkey, [email protected] E.B. Çil, Northwestern University, U.S.A., [email protected] F. Karaesmen, Koç University, Turkey, [email protected]

We study a class of queueing control problems involving commonly used control mechanisms such as admission control, pricing, and service rate control. Such controls are frequently employed in modeling service systems, telecommunications, and also in inventory systems modeled as make-to-stock queues. It is by now well established that in a number of such important problems, the optimal policy is structured and is described by a few parameters. From a design point of view, it is useful to understand how the optimal policy changes when the system parameters such as arrival or service rates, or the number of servers vary. We present a general framework to investigate the policy implications of the system parameters by using event-based dynamic programming. In this framework, the value function of the control problem is first represented by a number of common operators, and the effect of system parameters on the optimal policy is analyzed for each individual operator. We present this analysis for a set of frequently used operators. Whenever a queueing control problem can be modeled by these operators, the effects of system parameters on the optimal policy follows trivially from this analysis. This enables us to easily retrieve and generalize some of the existing results in the literature. We also present examples in a number of challenging recent problems.

Session TC08 - Queueing Models 2, Chair: Gómez-Corral in CZ 15

Bad luck when joining the shortest queue

Hans Blanc, Tilburg University, The Netherlands, [email protected]

A frequent observation in service systems with queues in parallel is that customers in other queues tend to be served faster than those in one’s own queue. We quantify the probability that the service of a tagged customer would have started earlier if that customer had joined another queue than the queue that was actually chosen, for exponential multiserver systems with queues in parallel in which customers join one of the shortest queues upon arrival and in which jockeying is not possible. We consider both symmetric

116 TC09 and asymmetric systems, and also systems with additional dedicated traffic to one or more servers. In symmetric systems, the probability of bad luck may well exceed 0.5 when there are three or more servers, but this only occurs if the load of the system is very close to 1. The approach of this probability to its heavy traffic limit is very steep, so that this limit, which is easily computable, will not be a good approximation for most values of the load. Asymmetry in the service rates tends to increase the probability of bad luck in light traffic, but to decrease it in moderate to heavy traffic. Dedicated background traffic decreases this probability.

On the correlation structure of Gaussian queues

A. Es-Saghouani, University of Amsterdam, The Netherlands, [email protected] M. Mandjes, University of Amsterdam, The Netherlands, [email protected]

We present results on the correlation structure of the workload process of a queue fed by Gaussian pro- cesses, such as fractional Brownian motion (fBm) and integrated Ornstein-Uhlenbeck (iOU) processes. The main question is: to what extent does the workload process inherit the correlation properties of the in- put process? We present an alternative definition of correlation that allows (in asymptotic regimes) explicit analysis. For the special cases of fBm and iOU we analyze the behavior of this metric under a many- sources scaling. Relying on (the generalized version of) Schilder’s theorem, we are able to characterize its decay. We observe that, popularly speaking, the correlation structure of the input process carries over to the workload process.

Negative Queue Length:Queues for Quantum Information

Hiroshi Toyoizumi, Waseda University, Japan, [email protected]

We study the detail of queues handling quantum communication, especially so-called quantum merging of qubits. In quantum merging, by using EPR pairs (entangled pairs of quantum particles), jobs may bring extra service capacity to the server, which can be stored in the system for future use. We will model this phenomena as queues that can have the negative queue length, and show the basic characteristics of these quantum queues.

On a batch Markovian arrival process subject to renewal catastrophes

A. Gómez-Corral, Complutense University of Madrid, Spain, [email protected]

The talk deals with a population of individuals that grows stochastically according to a batch Markovian arrival process and is subject to renewal generated geometric catastrophes. Our interest is in the semi- regenerative process describing the population size at arbitrary times. Towards this end, we study the underlying Markov renewal process whose embedded Markov chain (at post-catastrophe epochs) may be thought of as a Markov chain of GI/G/1-type, which is indeed amenable to be studied through its R- and G-measures, and a suitably defined Markov chain of M/G/1-type. We present tractable formulas for the equilibrium distribution of the population size and the distribution of the time to extinction for present individuals at post-catastrophe epochs, among other probabilistic descriptors.

Session TC09 - World Wide Web, Chair: Litvak in CZ 16

PageRank of Scale-Free Growing Networks

K. Avrachenkov, INRIA Sophia Antipolis, France, [email protected] D. Lebedev, R&D France Télécom, France, [email protected]

117 11 – Abstracts

PageRank is one of the principle criteria according to which Google ranks web pages. PageRank can be interpreted as a frequency of webpage visits by a random surfer, and thus it reflects the popularity of a webpage. In the present work we find an analytical expression for the expected PageRank value in a scale- free growing network model as a function of the age of the growing network and the age of a particular node. Then, we derive asymptotics that show that PageRank follows closely a power law in the middle range of its values. The exponent of the theoretical power law matches very well the value found from measurements of the World Wide Web. Finally, we provide a mathematical insight for the choice of the damping factor in PageRank definition.

Distances in Random Graphs

Remco van der Hofstad, Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, [email protected]

The structure of real networks, such as the WWW, Internet and various biological networks, is receiving increasing attention in both the mathematical and the networking community. Measurements have shown fascinating features, such as their ‘scale-free’ nature and the ‘small-world’ phenomenon. The small-world phenomenon states that typical distances in large complex networks are small, while a complex network is scale free when it has a power law degree sequence, i.e., when the number of nodes with degree proportional to k decays as an inverse power of k. It is difficult to model the topological structure of such networks, due to their high (and ever increasing) complexity. We shall discuss a few simple models for scale-free random graphs. We investigate the asymptotic law of the distances between typical pairs of vertices in the graph, and the structure of neighborhood exploration. For general complex networks, these distances are important features, since for example, the spread of diseases in a population is closely related to the topology of the social network describing the population. We hope that our findings prove relevant as well for the PageRank problem.

Stochastic equations behind Web ranking

Y. Volkovich, University of Twente, The Netherlands, [email protected] D. Donato, Yahoo! Research, Spain, [email protected] N. Litvak, University of Twente, The Netherlands, [email protected] W.R.W. Scheinhardt, University of Twente, The Netherlands, [email protected]

PageRank was introduced by Google as a popularity measure for Web pages. We study the probabilistic relation between PageRank and other parameters of the Web such as in-degree, out-degree and the fraction of pages with out-degree zero. We model this relation through stochastic equations inspired by the original definition of PageRank. We use the theory of regular variation to prove that the tail distributions of PageR- ank and in-degree follow power laws with the same exponent, which is in agreement with experiments. We also show that out-degree has a minor effect on the PageRank distribution.

A fair distribution of probability mass on the bow-tie graphs

Konstantin Avrachenkov, INRIA Sophia Antipolis, 2004 Route des Lucioles, 06902, France Nelly Litvak,University of Twente, P.O.Box 513 7500 AE Enschede The Netherlands Kim Son Pham,St.Petersburg State University, 35 University Prospect, 198504, Peterhof, Russia

Link-based popularity measures are an important tool used by search engines for ranking Web pages. We consider the Google PageRank that is defined as a stationary distribution of an ‘easily bored surfer’ random walk on a Web Graph. At each step, with probability c, such random walk starts afresh from the uniform distribution. The choice of the ‘damping factor’ c is not evident. The Google’s original choice c = 0.85

118 TD01 was a compromise between the true reflection of the Web structure and numerical efficiency. However, due to the bow-tie structure of the Web, the Markov random walk on the original Web Graph does not reflect the importance of pages because it absorbs in dead ends. We propose new criteria for choosing the damping factor based on the ergodic structure of the Web Graph. Specifically, we require that the core component receives a fair share of the PageRank mass. Using singular perturbation approach we conclude that the value c = 0.85 is too high and suggest that the damping factor should be chosen around 1/2. Our analytical results are confirmed by experiments on two large samples of the Web Graph.

Tuesday 4:45pm - 6:15pm

Session TD01 - Lévy Processes and Finance, Chair: Rieder / Bäuerle in CZ 4

Malliavin Calculus for Levy Processes and Applications in Finance

Evangelia Petrou, HU Berlin, Germany, [email protected]

Most of the literature on Malliavin Calculus and its applications to Finance focuses on markets that are generated by diffusion processes. Our main aim is to extend these results for markets that are generated by Levy processes. Proving a chaos expansion for Levy spaces we are able to define directional Malliavin derivatives and to reach a Clark-Ocone- Haussman (COH) formula. Furthermore we can define the adjoint( Skorohod integral) of the directional derivatives and we prove an integration by parts formula. The above theory will be used to explicitly estimate hedging strategies and the Greeks in markets generated by Levy processes.

Extremal behavior of stochastic integrals driven by regularly varying Lévy pro- cesses

H. Hult, Brown University, USA, [email protected] F. Lindskog, Royal Institute of Technology, Sweden, [email protected]

We study the extremal behavior of a stochastic integral driven by a multivariate Lévy process that is regu- larly varying with index α > 0. For predictable integrands with a finite (α + δ)-moment, for some δ > 0, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Lévy pro- cess and we determine its limit measure associated with regular variation on the space of right-continuous functions with left limits. The connection to popular models in finance is discussed.

Dependence properties and comparison results for Lévy processes

N. Bäuerle, University of Karlsruhe, Germany, [email protected] A. Müller, Heriot-Watt University Edinburgh, GB, [email protected] A. Blatter, University of Karlsruhe, Germany, [email protected]

We investigate dependence properties and comparison results for multidimensional Lévy processes. Asso- ciation, positive orthant dependence and positive supermodular dependence of Lévy processes are char- acterized in terms of the Lévy measure as well as in terms of the Lévy copula, a concept which has been introduced recently. As far as comparisons of Lévy processes are concerned we consider the supermodular and the concordance order and characterize them by orders of the Lévy measures and by orders of the Lévy copulas respectively. This generalizes recent results. Finally we show how these results can be applied in

119 11 – Abstracts

finance and insurance to obtain for example comparison statements for ruin times and probabilities as well as for option prices.

Session TD02 - Spatial Models 1, Chair: Foss / Stolyar in CZ 5

Large deviations of the interference in a wireless communication model

A.J. Ganesh, Microsoft Research, Cambridge, UK, [email protected] G.L. Torrisi, IAC, Rome, Italy, [email protected]

Interference from other users limits the capacity, and possibly the connectivity, of wireless networks. We consider a simple model of a wireless ad-hoc network where node locations are described by a homo- geneous Poisson point process, and node transmission powers are random. For light-tailed transmission powers we obtain a large deviation principle for the total interference, and relate the speed of the large devi- ation principle to the tail decay rate of the distribution of transmission power. For heavy-tailed transmission powers we provide asymptotic upper and lower bound for the tail of the interference distribution.

M/D/1/1 loss system with interference and applications to transmit-only sensor net- works

B. Błaszczyszyn, ENS/INRIA Paris, France and University of Wrocław, Poland, [email protected] B. Radunovic´, Microsoft Research, Cambridge, UK, [email protected]

We propose and analyze a probabilistic model of packet reception in the steady state regime of a non-slotted wireless communication channel. It is an extension of the classical M/D/1/1 Erlang’s loss model where the interference created by different packet emissions is introduced by means of a shot-noise process. More precisely, we assume that a given packet is admitted by the single receiver if this latter is idle at the packet arrival epoch and successfully received if, in addition, its signal-to-interference-and-noise ratio averaged over the reception period is large enough. As the main result we prove an analog of the Erlang’s formula for the ergodic rate of the successfully received packets. Our work is motivated by some applications to transmit-only sensor networks.

Local and global properties of random directed trees

Rahul Roy, Indian Statistical Institute, New Delhi, [email protected]

Random directed trees are constructed in space as well as in the unit cube. For the spanning trees in space we study the geometric structure – in particular, when does the construction give us a connected graph, and when it gives a collection of disjoint trees. For the construction in the unit cube, we study the asymptotic properties of the graph as the number of vertices goes to infinity.

Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models

M. Lelarge, INRIA-ENS, France, [email protected] David Aldous C. Bordenave, University of California at Berkeley, United States, [email protected]

We study the relation between the minimal spanning tree (MST) on many random points and the “near- minimal” tree which is optimal subject to the constraint that a proportion δ of its edges must be different

120 TD03 from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as 1 + 2(δ2). We prove this scaling result in the model of the lattice with random edge-lengths and in the 2-dimensional Euclidean model.

Session TD03 - Asymptotics, Chair: Blanchet / Mikosch in CZ 10

Towards Efficient Simulation of Long Delays for Heavy-tailed Multiserver Queues

Jose Blanchet, [email protected] Peter Glynn, [email protected]

We consider the question of computing via simulation the likelihood of observing at least one customer that experiences a long delay within a busy period in a multiserver queue with heavy-tailed input. We shall discuss a class of algorithms that can be argued heuristically to achieve asymptotic optimality for multiserver queues with regularly varying input throughout a wide range of stability conditions. In the case of two server queues, our algorithms can be rigorously proved to achieve asymptotic optimality if the traffic intensity is not equal to 1/2. In the single-server queue case, the algorithm is particularly easy to implement and produces a new asymptotically optimal procedure for estimating the tail of the steady-state waiting time.

Large deviations for point processes based on stationary sequences with heavy tails

H. Hult, Brown University, United States, [email protected] G. Samorodnitsky, Cornell University, United States, [email protected]

A stationary sequence of random variables with regularly varying tails is considered. For this sequence it is possible that large values arrive in clusters. That is, there may be many large values in a relatively short period of time. The aim is to give a detailed description of the occurrence of large values, not only the size of clusters but also within a single cluster. To do this a point process based on appropriately scaled points of the stationary sequence is constructed. A limiting measure, on the space of point measures, describes the joint limiting behavior of all the large values of the sequence. From this limiting result one can proceed and obtain the functional large deviation for its partial sums, ruin probabilities, etc. Examples include, linear processes, random coefficient ARMA processes, and solutions to stochastic recurrence equations.

Rare event simulation methodology for perpetuities

Jose Blanchet, Harvard University, USA, [email protected] Bert Zwart, Georgia Institute of Technology, USA, [email protected]

A perpetuity is the present value of a bond in the presence of possible random interest rates and premiums, in a discrete or continuous-time setting. Due to the nonlinear structure of a perpetuity, its tail behavior can be heavy tailed, even if all input random variable are light-tailed. Large deviations estimates (see Goldie (1991) among many others) among many others) yield power tail behavior of the form Ct−κ , where κ is easily computable, but C is not. Since the precise convergence rate towards this asymptote is also not clear, it is desirable to have efficient simulation techniques available. The goal of this talk is to present several of such techniques. We consider several different cases and present simulation algorithms which are asymptotically optimal. A typical feature of these algorithms is that they are state dependent, in contrast with standard importance sampling techniques.

121 11 – Abstracts

Boundary behavior and stability for multidimensional reflected diffusions with state- dependent jumps in the positive orthant

Ravi R. Mazumdar, Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Canada

Reflected diffusions with jumps occur in many financial as well as queueing network models. There has been much work by Williams and Harrison on Semi-martingale Reflected Brownian Motion (SRBM) (ICM 1998) and the boundary characterization of such processes. However there are few results for reflected dif- fusions with jumps with non-constant drifts and diffusion coefficients and state-dependent reflections. One of the difficulties is that the associated Skorohod problem does not have a Lipschitz continuous reflection map. In the talk I will present 2 new sets of results for diffusions with jumps and state-dependent oblique reflections at the boundary. The first is a characterization of the boundary behavior in terms of the local time and a result on the support of the distribution on the boundary surfaces. In particular the results gen- eralize some earlier results due to Taylor and Williams (PTRF 1993) and Guillemin and the speaker (Appl. Math. Opt. 1996). In the second part, we exploit the boundary characterization to obtain the necessary and sufficient conditions for the existence of product-form stationary distributions and sufficient conditions for their existence. I will conclude with the general case with state-dependent reflections in a so-called monotone case. Based on joint work with Dr. Fabrice Guillemin (CNET, France) and Francisco Piera (University of Chile).

Session TD04 - Large Deviations 1, Chair: Majewski in CZ 11

Large deviations of multitype queues

A. Dabrowski, University of Ottawa, Canada J. Lee, Yeungnam University, Republic of Korea, [email protected] D.R. McDonald, University of Ottawa, Canada, [email protected]

Consider a multitype queue where queued customers are served in their order of arrival at a rate which depends on the customer type. We derive sharp asymptotics of the steady state distribution of the queue size by finding an appropriate harmonic function h for the Markov additive process. The associated h- transform produces a change of measure that increases the arrival rate of customers and decreases the departure rate thus making large deviations common. Using the change of measure we calculate the sharp asymptotics of the probability that the total number of customers in the queue reaches a high level before emptying and we also obtain a description of the queues when this large deviation occurs. We present numerical results obtained via our asymptotic expressions and compare those results with the simulation results.

Limiting distribution of the phase when level passes a threshold

Aziz Khanchi, University of Ottawa, Canada, [email protected] David R. McDonald, University of Ottawa, Canada, [email protected]

Consider a network with n nodes. We provide the limiting distribution of the remaining nodes at the moment the first node crosses a threshold which tends to infinity. The performance of the system is studied by twisting the associated Markov additive process using a suitable harmonic function. The current results are extended to the cases where the Markovian component of the twisted Markov additive process is not positive recurrent. We apply our results to a Modified Jackson Network, where the second node helps the first one whenever the second node is idle. The existence of the limiting distribution for this network is rather surprising since the large deviation bridge path avoids emptying the second node.

122 TD05

Large Deviations of Generalized Jackson Networks

Silke Meiner

Investigating generalized Jackson networks (not necessarily exponential iid inter event times and Bernoulli routing independent of everything else) I develop the rate function for the sample path Large Deviation of queue sizes. The rate function is identified on tubes around affine functions. My approach follows Ignatiouk-Robert (2000) and main stages in the argumentation are: an LD upper bound with the help of an exponential change of measure, the solution of an optimization problem in the upper bound and the interpretation of the optimizer. Associated with the optimizer is a change of measure under which the network can be divided into two subnetworks: one in equilibrium the other behaving like a free process. Tightness of the upper bound follows from properties of the network under the change of measure corre- sponding to the optimizer found in the upper bound.

Functional asymptotics of a simple queueing network with short- and long-range dependent input

Kurt Majewski, Siemens AG., Corp. Tech., München, Germany, [email protected]

We consider an exemplary queueing network with two nodes providing infinite buffer space and non-idling fifo service at constant rate. Customers of a constant rate traffic have to visit both queues whereas the customers of a random cross traffic just visit the first queue before leaving. The queue length behavior of the second queue is a measure for the delay variation induced on the constant rate traffic by sharing the first buffer with the random cross traffic. Functional weak heavy traffic (Peterson 1991), large deviation (Ramanan and Dupuis 1995), and moderate deviation (Majewski 2006) asymptotics of this queue length behavior have been developed for suitably scaled renewal cross traffic. We extend these three types of asymtptotics to certain appropriately scaled sequences of Markovian and long-range dependent cross traffic. The weak heavy traffic limits of the queue length behavior for the three types of cross traffic processes can be characterized as twice reflected Brownian motion, integrated Ornstein-Uhlenbeck process, and fractional Brownian motion, respectively. The minimizing paths leading to a large buffer content in the second queue in the large and moderate deviation principles can be characterized explicitly and calculated numerically. They give insight into the transition from short- to long-range dependent input in queueing networks.

Session TD05 - Computer-communications, Chair: Hwang in CZ 12

Asymptotic results in some random infinite urn models

P. Robert, INRIA-Rocquencourt, France, [email protected] F. Simatos, INRIA-Rocquencourt, France, [email protected]

Motivated by the problem of diffusion of information in peer to peer networks, a stochastic model for which N balls are thrown in an infinite number of urns is studied. The probability pi that a ball is in urn i is a random number, which makes the balls dependent, and the asymptotic behaviors of the index of the first empty urn as well as the index of the last occupied urn are investigated.

Connectivity probability of one-dimensional ad hoc wireless networks with mini- mum hop path

Nelson Antunes, CEMAT/University of Algarve, Faro, Portugal, [email protected]

123 11 – Abstracts

Gonçalo Jacinto, CIMA-UE/University of Évora, Évora, Portugal, [email protected] António Pacheco, CEMAT/Instituto Superior Técnico, Lisboa, Portugal, [email protected]

In an ad hoc multi-hop network, the functionality of the network critically depends on the connectivity properties of the network. These properties depend on the number of nodes, their transmission ranges and spatial distribution. This work investigates the connectivity probability in an one-dimensional ad hoc network from a different viewpoint: minimum hop path. We consider a source and a destination node and a finite set of relay nodes uniformly distributed between them. The exact probability distribution of the minimum number of hops (M) between the source and destination is derived from the density location of relay nodes in the minimum hop path. The analysis is based on the Poissonification technique. The probability that the source and destination pair is connected (provided by Ghsemi and Nadser-Esfahani (2006)) can be obtained by summing the probability masses for each possible number of nodes in the minimum hop path. We provide an analytical expression for network design that enables the determination of the required number (resp. transmission range) of relay nodes given a desired probability of connectivity with the minimum value of M. Furthermore, we also study the distribution of the minimum hop path duration and the mean length of connectivity periods for the different values of M. Finally, some numerical results are presented.

Design and Analysis of a Packet Scheduler Exploiting the Multiuser Diversity for Performance Enhancement

Gang Uk Hwang, Korea Advanced Institute of Science and Technology (KAIST), Republic of Korea, [email protected] Fumio Ishizaki, Nanzan University, Japan, [email protected]

In this paper, we consider a new packet scheduler exploiting the multiuser diversity in a wireless network with an AMC (Adaptive Modulation and Coding) scheme, and propose an analytic model based on the effective bandwidth theory to investigate the joint effect of the AMC scheme and the packet scheduler on system performance. Based on our analytic results as well as numerical results, we show how to design the packet scheduler to improve the spectral efficiency of the wireless channel.

Session TD06 - Applied Probability, Chair: Hagwood in CZ 13

Stochastic modeling of a dark matter experiment

Kevin Coakley, National Institute of Standards and Technology, [email protected]

In a proposed experiment mini-CLEAN (Cryogenic Low Energy Astrophysics with Noble gases), Weakly Interacting Massive Particles (WIMPs) might be detected based on scintillation light produced when a WIMP deposits energy in a scintillator by nuclear recoil scattering. The WIMP is a candidate particle to explain dark matter in the universe. In this experiment, any WIMP signal would be very weak compared to a very intense background that produces scintillation light by electronic recoil scattering. I will discuss stochastic models of observed pulses produced by both WIMP events and background events, and spatial and temporal methods to discriminate WIMP events from background events.

Stochastic Modeling in Gene Expression Measurements: Accounting for Both Back- ground Error and Instrument Saturation

Z.Q. John Lu, National Institute of Standards and Technology, [email protected]

124 TD07

There are multiple ways of measuring expressions of thousands of genes simultaneously through measuring the amount of mRNAs by building probes using either the full complementary sequences or using sets of probes pairs based on short subsequences (e.g. Affymetrix arrays). Sources of error include both the non-specific hybridization from using short oligonucleotide arrays to scanner performance limitations due to using the two-color fluorescent dye effects, optical noises, and saturation at high intensity readings. NIST chemists have performed a careful study of two brand scanners over a long period, from which we were able to characterize the scanner effects in the measurement process. In this paper, I describe some statistical issues in fitting a general nonlinear measurement error model for microarray gene expression measurements, which can account for both the background error and instrument saturation, while allowing the signal to vary at a large exponential or Langmuir hyperbolic dynamic range.

Stochastic Modeling of a Carbon Nanotube in an Electrical Field

Charles Hagwood, National Institute of Standards and Technology, [email protected]

We want to classify diameter selected carbon nanotubes by length. In this talk a stochastic differential equation is used to describe the motion of nanotubes (or nanowires) undergoing Brownian rotation and translation in an electric field.

Allocation Models and heuristics for the outsourcing of repairs for a dynamic war- ranty population

L. Ding, The Management School, Edinburgh University, Edinburgh EH8 9JY, UK K.D. Glazebrook, Department of Management Science, Lancaster University, LA1 4YX C. Kirkbride, Department of Management Science, Lancaster University, LA1 4YX

We consider a scenario in which a large equipment manufacturer wishes to outsource the work involved in repairing purchased goods while under warranty. Several external service vendors are available for this work. We develop models and analyses to support decisions concerning how responsibility for the warranty population should be divided between them. These also allow the manufacturer to resolve related questions concerning, for example, whether the server capacities of the contracted vendors are sufficient to deliver an effective post sales service. Static allocation models yield information concerning the proportions of the warranty population for which the vendors should be responsible overall. Dynamic allocation models enable consideration of how such overall workloads might be delivered to the vendors over time in a way which avoids excessive variability in the repair burden. We apply DP policy improvement to develop an effective dynamic allocation heuristic. This is evaluated numerically and is also used as a yardstick to assess simpler heuristics suggested by statis models. A dynamic greedy allocation heuristic is found to perform well. It is clear that dividing the workload equally among vendors with different service capacities can lead to serious losses.

Session TD07 - Optimal Stopping 1, Chair: Gnedin in CZ 14

Maximizing the expected duration time related to Poisson arrival

A. Kurushima, Tokyo University of Science, Japan, [email protected] K. Ano, Institute of Applied Mathematics, Japan, [email protected]

Here we consider the following optimal stopping problem. Suppose the object arrives sequentially ac- cording to Poisson process with unknown intensity having Gamma prior density G(r, 1/a), where a is nonnegative. Let the time horizon be (0, T ] of finite length. Suppose the objects is rankable without equal

125 11 – Abstracts rank and we observe only the relatively rank arrived so far. We want to maximize the expected duration time owing the relatively best object, so that our problem is regarded as a variation of the secretary problem. For each r = 1,2 and 3, we derive the optimal stopping time. It is a threshold type which can be described ∗ = { ∈ [ (r)∗ ]: = } { = } as τr min t si , T Xi 1 , where the event Xi 1 means that ith object is the relatively (r)∗ best object and si is a unique root of a certain equation for each r. We show the threshold sequence of { (r)∗} (r)∗ [ − 2 − ]+ → ∞ si i≥1 is nonincreasing and the asymptotic value si goes to (T a)/e a as i for all r = 1,2 and 3. Furthermore we shall investigate the nature of the optimal stopping time for the general case with a natural number r.

A linear programming analysis of finite horizon optimal stopping problems

K. Helmes, Humboldt University of Berlin, Germany, [email protected]

We numerically analyze finite horizon optimal stopping problems by exploiting an infinite dimensional lin- ear programming formulation of such problems. We approximate the infinite dimensional linear programs (LP) by particular finite dimensional LPs. The finite dimensional linear programs to be solved are based on approximations of moment sequences of finite measures with compact support. The finite dimensional LPs provide bounds on the optimal value as well as accurate information about the optimal stopping rules.

Optimal stopping with a rank-dependent loss

A.V. Gnedin, Utrecht University, The Netherlands, [email protected]

For τ a stopping policy adapted to a sequence of n iid observations, we define the loss to be Eq(Rτ ), where Rτ is the rank of the τth observation, and q is a nondecreasing function of the rank. This setting covers both the best choice problem with q(r) = 1(r > 1), and Robbins’ problem with q(r) = r. We show that for large n the optimal stopping problem has a scaling limit associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal policy like complete history dependence.

Optimal Stopping of ”Seasonal” Observations

E.L. Presman, Central Economics & Mathematics Inst. of Russian Academy of Sci., Russia, [email protected] I.M. Sonin, Univ. of North Carolina at Charlotte, USA, [email protected]

Our talk has two main goals: first, to describe a new class of optimal stopping problems for which the solutions can be found in a finite number of steps, and in many cases in an explicit form, and second, to demonstrate the potential of the Elimination algorithm (EA) developed by one of the authors earlier. We present this algorithm in a modified form for an arbitrary state space.

Xn s Let B = {1,..., m}, a statistician observes a random sequence Zn = (Xn, Yn ), n ≥ 0, where (Yn )n≥0 for s each s ∈ B is an i.i.d. random sequence with distribution F , and (Xn)n≥0 is a Markov chain independent S   of (Yn)n≥0, taking values in {e} B with transition probabilities P Xn+1 = j |Xn = i = pi j for i, j ∈  = | =  = =
≥ B, P Xn+1 e Xn e 1. The matrix P pi j i, j∈B is a strictly substochastic matrix, i.e. pi j 0 X and 0 < pi j < 1 for all i ∈ B, and e is an absorbing state. The goal is to maximize Eg(Zτ ) over j∈B all possible stopping times τ, g( j, y) is a reward function. We prove that there exists a threshold vector c = (c1,..., cm) such that S = {( j, y) : j ∈ B, g( j, y) ≥ c j } is an optimal stopping set. We give a sufficient condition when the EA solves the problem of finding the value function and the vector c in a finite number of steps. The work of E.L.Presman was supported by RFBR grant 07-01-00541.

126 TD08

Session TD08 - Queueing in Telecom, Chair: Tijms in CZ 15

A spectral method for a BMAP/D/c queue with synchronous servers

S. Nishimura, Tokyo University of Science, [email protected] N. Yatomi, Tokyo University of Science, [email protected]

A batch Markovian arrival process (BMAP) is great flexible and its queueing models are analytically tractable for a modeling of IP traffic with a self-similar nature. Since IP traffic is continuous and data are transmitted by synchronous multiple servers, we consider a BMAP/D/c queue with synchronous servers, where the arrival process to the system is a continuous-time BMAP defined by M × M rate matrices and the beginnings of c servers with the constant service time h are synchronized as 0, h, 2h,.... We propose a spectral method for calculations of the stationary probability of the number of customers in the system at synchronized epochs. A spectral method is fit to a BMAP with a large maximum batch size as IP traffic. First, by the modified Durand-Kerner method, the boundary vector with the size Mc is derived from the M zeros of the c nonlinear equations, independently. Next, by the inverse discrete Fast Fourier Transform (FFT) method for the vector probability generating function, the stationary probability is calculated. In a numerical example, the stationary probability and its tail probability are efficiently calculated for the BMAP, which is estimated from real IP traffic and has a good fitness to traffic. Effects of the number of servers on the stationary probabilities are discussed.

Differential Traffic Theory

J.M. Garcia, LAAS-CNRS, France, [email protected] O. Brun, LAAS-CNRS, France, [email protected]

Network operators have a growing need for efficient network simulation and planning tools (topology design, capacity planning, traffic engineering). One key component of such tools is the performance evalu- ation kernel. Given the limits of pure event-driven simulation methods, differential traffic modeling appears as an efficient alternative for performance evaluation of large telecommunication systems. These models are based on approximations of the exact differential traffic equations obtained using the notion of implicit transient traffic. These models, have been introduced in commercial network planning tools (NetQUAD, NEST) and are extensively used for telephone and IP-MPLS networks. They proved their efficiency and accuracy for large scale networks. We will present this modeling framework for circuit switched systems, multirate loss networks and packet-switched networks. Extension to an innovating hybrid simulation tech- nique will be developed.

Moments of the duration of busy periods in oscillating queueing systems

A. Pacheco, CEMAT/Instituto Superior Técnico, Lisboa, Portugal, [email protected] H. Ribeiro, CEMAT/Instituto Politécnico de Leiria, Leiria, Portugal, [email protected]

We compute moments of the duration of busy periods in oscillating M/G/1 systems with finite capacity and batch arrivals, for which the service time distribution oscillates between two forms according to the evolution of the number of customers in the system. We work with an extended definition of busy period (that may be) initiated by multiple customers and derive an efficient recursive algorithm to compute mo- ments of the duration of a busy period starting with an arbitrary number of customers in the system. We derive a recursive algorithm on the capacity of the system that is based on the fact that the duration of a busy period initiated by multiple customers in a system of some fixed capacity can be expressed in terms of the durations of busy periods of systems with smaller or equal capacities and initiated by a single customer.

127 11 – Abstracts

A quick approximation for the state probabilities and waiting-time probabilities in the M/D/1 queue

H.C. Tijms, Vrije University Amsterdam, The Netherlands, [email protected]

We present amazingly accurate approximations to the state probabilities and the waiting-time probabil- ities in the M/D/1 queue using a two-phase process with "branching probabilities" outside (0,1). The approximations are in the form of explicit expressions involving geometric and exponential terms. The approximations extend to the finite-capacity model

Session TD09 - Engineering, Chair: Günalay in CZ 16

Analysis of a Probabilistic Key Management Scheme for Vehicular Communications

E. van den Berg, Telcordia Technologies, USA, [email protected]

Vehicular communication networks have drawn much attention recently, notably for their promise of en- hancing vehicle safety. By broadcasting warning messages, vehicles can notify each other of impending hazards and recent accidents. Given the criticality of the safety applications, these messages must be en- crypted to enable secure communications. A simple probabilistic scheme exist for management of the required cryptographic keys. The scheme has been studied extensively in the context of broadcast en- cryption. Of particular concern for practical deployment vehicular communications networks is vehicle privacy: it is critical to avoid messages to and from vehicles being used to identify or track vehicles. We describe a new method for a vehicle to select cryptographic certificates / keys which can be used with the probabilistic key management scheme. The method allows a vehicle to use keys which it has in common with other vehicles in its direct neighborhood, thus increasing the privacy/anonymity of the vehicles. We analyze the performance of the method using random graphs and Markov chains.

Comparison of the effect of various forecasting methods on bullwhip effect in supply chain

Reza Zanjirani Farahani, Industrial Engineering Department, Amirkabir University of Technology, Tehran, Iran Mehdi Najafi, Industrial Engineering Department, Amirkabir University of Technology, Tehran, Iran Nasrin Asgari, Supply Chain Management Research Group, Tehran, Iran

Nowadays, enterprises in supply chain are attempting to increase their efficiency and effectiveness as well as supply chain’s. Therefore, identification and consideration of factors which prevent enterprises to at- tain their expected effectiveness are very important. Inasmuch as one of these main factors is bullwhip effect, being aware of its reasons help the enterprises to decrease the severity of bullwhip effect by opting proper decisions. Now that forecasting method is one of the most important factors in increasing or de- creasing of bullwhip effect, this paper consider and compromise the effect of various forecasting methods on bullwhip effect. In fact in this paper, the effect of various forecasting methods such as Moving Average and Exponential Smoothing as well as Regression in bringing about bullwhip effect in a four layer supply chain -including: retailer, wholesaler, manufacturer and supplier- is considered. Then, the bullwhip effect measure is utilized to compromise the ineffectiveness of various forecasting methods. Owing to this, we generate random demands in two states including constant and increasing trend of demands. Eventually, we ranked the forecasting methods in two state of demand and utilize statistical method to certain about significance of difference among effect of various methods. Keywords: Bullwhip Effect, Supply chain, Forecasting, Exponential Smoothing, Moving average, Simula- tion.

128 WA01

Using Conditional Probabilities for Automatic New Topic Identification of Search Engine Transaction Logs

Seda Ozmutlu, Industrial Engineering Department Uludag University, Bursa, Turkey, [email protected] Huseyin C. Ozmutlu, Industrial Engineering Department Uludag University, Bursa, Turkey, [email protected] Buket Buyuk, Industrial Engineering Department Uludag University, Bursa, Turkey, [email protected]

One of the most important dimensions of search engine user information seeking behavior and search engine research is content-based behavior. The purpose of this study is to perform automatic new topic identification in search engine transaction logs using conditional probabilities of new topic arrivals. Sample data logs from FAST and Excite are used in the study. Conditional probabilities of new topic arrivals and topic continuations given query category are used to estimate new topic arrivals. The findings of this study show that the conditional probability approach yielded favorable results in terms of the performance measures used for evaluating automatic new topic identification algorithms.

A queueing analysis of a hybrid manufacturing system with new and refurbished items

Y. Günalay, Bilkent University, Turkey, [email protected] D. Gupta, University of Minnesota, USA, [email protected]

Consider a direct-to-market manufacturer which sells both new and refurbished versions of the same item. The firm uses make-to-stock policy for both product. Server performances are related through demand and perceived product substitutability by customers. The system is modeled as two interacting queues: fist as a finite population M/M/1 queue and the second as a Quasi-Birth-Death process. The system is analyzed for different product characteristics and the best base-stock levels are calculated using exact or approximate numerical methods.

Wednesday 8:30am - 10:00am

Session WA01 - Queues and Lévy Processes, Chair: Kella in CZ 4

The M/G/1 queue with quasi-restricted accessibility

David Perry, University of Haifa, Israel, [email protected]

We consider single-server queues of the M/G/1 type with a special kind of partial customer rejection called quasi-restricted accessibility (QRA). Under QRA, the actual service time assigned to an arriving customer depends on his service requirement, say x, the current workload, say w, and a prespecified threshold b. If x + w ≤ b the customer is fully served. If w ≤< w + x, the customer receives service time b − w + (w + x − b) f for some random number f ∈ [0, 1], while if w > b the actual service time is the fraction f x of the requirement. The random fractions are assumed to be i.i.d. We derive the following measures for the M/G/1 and the G/M/1 models: steady-state law, busy period analysis and cycle maximum. Joint work with Onno Boxma, Wolfgang Stadje, and Shelley Zacks.

129 11 – Abstracts

Lévy correlation cascades

Iddo Eliazar, Holon Institute of Technology and Tel Aviv University, Israel, [email protected]

Random processes with heavy-tailed distributions have infinite variances. Thus, their correlation-structures cannot be investigated via their auto-covariance functions - which are undefined. We consider a large class of random processes driven by non-Gaussian Levy noises. Exploiting the underlying Poisson-superposition structure of the driving noises, we introduce a cascade of Poissonian Correlation Functions which character- ize the process-distribution of the random processes under consideration. Specific examples include: Levy motions; Levy-driven Ornstein-Uhlenbeck processes; Levy-driven Moving-Average processes; Fractional Levy motions; Fractional Levy noises; and, general Shot Noise processes. The "Poissonian methodology" developed enables a precise quantitative analysis of Levy-driven random processes displaying, simultane- ously, the Noah effect (heavy-tailed distributions) and the Joseph effect (long-range dependence). Joint work with Joseph Klafter.

Superposition of renewal processes and an application to multi-server queues

Wolfgang Stadje, University of Osnabrück, Germany, [email protected]

The aim of this talk is to compare the waiting times of customers in multiple-server queues, where the idle times are removed, with different numbers of servers. For this purpose we develop some results regarding the vector valued marked point process whose points are arrival epochs of the superposition of renewal processes with different continuous inter-arrival distribution and the marks are the vectors of forward recurrence times of the various renewal processes at these arrival epochs. Joint work with Offer Kella.

Lévy driven fluid queues with negatively correlated service interruptions and busy periods

Offer Kella, The Hebrew University of Jerusalem, Israel, [email protected]

We consider a Lévy driven fluid queue which evolves as follows. Given some initial positive state, the process behaves like a Levy process with no negative jumps having a negative mean. At the instant when the process hits the origin the hitting time is compared with some independent exponential random variable. If the hitting time is smaller, then the station becomes idle for a time period which is a constant multiple of this hitting time. During this time the process behaves like a subordinator (input and no output). If the hitting time is larger then the system remains empty for an exponential time period at the end of which there is a jump of random length. We study the stability of this process and its steady state characteristics. The analysis involves the study of the iterations of some concave function and their limit. Joint work with Onno Boxma and Michel Mandjes.

Session WA02 - Random Graphs, Chair: Norros in CZ 5

Two diameter results in the configuration model

Gerard Hooghiemstra, Delft University of Technology, The Netherlands, [email protected]

This talk(based on joint work) gives two results on the diameter for the configuration model.

130 WA02

Denote by D a copy of the i.i.d. degree sequence D1, D2,..., DN , from which we built the random graph. We assume that D has a power law with exponent τ. Then the first result is:

For τ > 2, f1 + f2 = Pr(D = 1) + Pr(D = 2) > 0 and f1 < 1 we will show that α log N, for some constant α, is a lower bound for the diameter. This result is most interesting for τ ∈ (2, 3), because we see that in this case the diameter and the typical distance are of a different order of magnitude.

A second result gives an upper bound on the diameter for τ ∈ (2, 3). For f1 + f2 = 0 and τ ∈ (2, 3), a constant times log log N is an upper bound for the diameter. If time allows I will sketch a proof of these two results.

A local limit theorem for the critical random graph

T. Müller, EURANDOM / Technische Universiteit Eindhoven, the Netherlands, [email protected] R. van der Hofstad, EURANDOM / Technische Universiteit Eindhoven, the Netherlands, [email protected] W. Kager, EURANDOM / Technische Universiteit Eindhoven, the Netherlands, [email protected] [email protected]

n
The Erdos-Rényi˝ random graph G(n, p) is constructed by taking n vertices and including each of the 2 possible edges independently with probability p. In their seminal work around 1960 Erdos˝ and Rényi = c already showed that if p n (for c a fixed constant) then, as n tends to infinity, the largest component is of order ln n with high probability if c < 1 and of order n with high probability if c > 1. Here we study the := 1 + λ ∈ Erdos-Rényi˝ random graph inside the “critical window”, when p n 4 (where λ R is an arbitrary n 3 constant). Previous work by various authors has shown that in this range the largest component is of order 2 2 n 3 with high probability, and that the number of vertices in the largest component (divided by n 3 ) has a continuous limiting distribution. We will derive an expression for the probability density function of the limiting distribution of the largest component, and a useful differential equation that the probability density function satisfies.

Percolation on the 2D Hamming graph: The supercritical phase

M.J. Luczak, London School of Economics, United Kingdom, [email protected] R. van der Hofstad, Eindhoven University of Technology, The Netherlands, [email protected]

We study random subgraphs of the graph H(2, n), which is the Cartesian product of two complete graphs = 1+ ∈ on n vertices. Every edge is independently present with probability p 2(n−1) for some  R. In a sequence of three papers, Borgs, Chayes, van der Hofstad, Slade and Spencer studied phase transitions in a large class of random graphs including H(2, n), obtaining precise estimates for the maximum size of a connected component in the subcritical and critical regimes, as well as upper bounds in the supercritical phase. No general lower bounds on the largest supercritical component exist; indeed, it is thought that the geometry of a given graph will play a crucial role in lower bounding its giant component. Here, we complement the results of Borgs et al. for the supercritical phase in the Hamming graph H(2, n). We prove that, when   (log n)1/3n−2/3, then the largest connected component has size close to 2n2 with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of p really are supercritical. Except for the factor (log n)1/3, this identifies the size of the largest connected component all the way down to the critical p window. This is another step towards understanding phase transitions in graphs "with geometry" (not the complete graph).

Robustness against core losses in a power-law random graph

I. Norros, VTT, Finland, ilkka.norros@vtt.fi H. Reittu, VTT, Finland, hannu.reittu@vtt.fi

131 11 – Abstracts

We work on the conditionally Poissonian random graph model (Adv.Appl.Prob. 38:1) and focus on the case of capacities (expected degrees) with finite mean and infinite variance. The typical distances within the largest component are then 2(log log N), where N is the number of vertices. The shortest paths tend to have an up-down shape (‘up’ referring to higher capacity) and to travel through the top of the ‘core’ consisting of vertices whose capacities exceed a certain level, growing with N. It is then interesting to study how the shortest paths change when a top part of the core is cut away. We consider the core as consisting of 2(log log N) ‘tiers’ and show that when a fixed number of top tiers is removed, the shortest paths increase at most by a constant number of hops when N → ∞. The main idea is that each tier has as subgraph a dense classical random graph, whose diameter is given by Bollobás’ well-known result. When the whole core is removed, however, the nature of the graph changes more deeply, and the distances become 2((log N)/`(N)), where `(N) = o(log log log N). It is also possible to follow the gradual change of path lengths between these extremes.

Session WA03 - Simulation 2, Chair: Grassmann in CZ 10

Zero-Variance Importance Sampling Estimators for Markov Process Expectations

H.P. Awad, University of Miami, Florida, [email protected] P.W. Glynn, Stanford University, California, [email protected] R.Y. Rubinstein, Technion – Israel Institute of Technology, Israel, [email protected]

We study the structure of zero-variance importance sampling estimators for expectations of functionals of Markov processes. For a class of expectations that can be characterized as solutions to linear systems, we show that a zero-variance estimator can be constructed by using an importance distribution that preserves the Markovian nature of the underlying process. This suggests that good practical importance sampling distributions can be found by searching within the class of Markovian probability distributions. The class of expectations considered includes as particular cases, among others: the mean time until hitting a rare set, the expected cumulative discounted reward until hitting a set, the mean duration of an excursion, the transient (deterministic horizon) expectation of a discounted final payoff, and steady-state expectations.

Fluid Limit for Hybrid MCMC samplers

G. Fort, LTCI, CNRS/GET, 46 rue Barrault, 75634 Paris Cedex 13, France, [email protected] E. Moulines, LTCI, CNRS/GET, 46 rue Barrault, 75634 Paris Cedex 13, France, [email protected] P. Priouret, LPMA, Université P.M.Curie, Boîte courrier 188, 75252 Paris Cedex 05, France, [email protected]

Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation, and optimization. In Fort et al. (2006), these techniques have been extended to a class of skip-free Markov chains : by scaling time, space, and the initial condition by a large constant, a fluid approximation is obtained that describes the Markov dynamic in the macroscopic form. These extensions were then applied to characterize the stability of some Markov Chain Monte Carlo (MCMC) samplers, namely some Metropolis-Hastings algorithms. MCMC algorithms depend on many design parameters that need to be finely tuned in order for the al- gorithm to converge efficiently. In this talk, we will caracterize the fluid limit of some hybrid MCMC samplers. We will then discuss and show how the fluid limit yields information for a partial automation of the choice of the design parameters. G. Fort, S. Meyn, E. Moulines and P. Priouret, ODE methods for skip-free Markov Chains stability with applications to MCMC, arXiv math.PR/06072800, 2006.

132 WA04

Applied Probability Contribution to the Burn-in Phase in Equilibrium Simulation

W.K. Grassmann, University of Saskatchewan, Canada, [email protected] S. Sethi, University of Saskatchewan, Canada, [email protected]

In steady-state simulation, one typically uses time averages to estimate equilibrium expectations. Since for Markovian systems with up to a million states, means and variances of time averages can be found by using randomization, applied probability can make and should make a contribution to this discussion. In our talk, we will, in particular, discuss the issue of the burn-in phase in simulation. Through experimentation, we found that unless the initial state has a very low probability, using a burn-in phase reduces the mean squared deviation between the time average and the equilibrium expectation, and one should therefore use a burn-in phase only to obtain a typical initial state. These experiments are then supported by logical arguments.

Ordering of time correlations in stationary Markov processes and queueing net- works

R. Szekli, University of Wroclaw, Poland, [email protected] H. Daduna, Uni Hamburg, Germany, [email protected]

We compare dependence in stochastically monotone Markov processes with partially ordered Polish state spaces using the concordance and supermodular orders. We show necessary and sufficient conditions for the concordance order in terms of the one step transition probabilities for discrete time processes and in terms of the corresponding infinitesimal generators for continuous time processes. We indicate relations of dependence orderings to the asymptotic variance reduction effect in Monte Carlo Markov Chains (MCMC), to Cheeger’s constants, and to positive dependence for Markov processes. Further we show for X an ergodic Jackson network processes with a routing matrix R and eX another Jackson network processes having the = [ ] same service intensities but with the routing matrix Re eri j such that the traffic solutions of the traffic equation for R and for Re coincide, that for arbitrary real functions f, g

X eX f,g ( f, Q g)π − ( f, Q g)π = λ/ξ0 Eπ (tr(W (Xt ) · diag(ξ) · (R − Re))), where diag(ξ) = (δi, j · ξi : 0 ≤ i, j ≤ J)) is the diagonal matrix with entries of the probability solution of f,g the traffic equation, and W (n) = [gi (n) f j (n)]i, j=0,1,...,J . Using this fact we compare time correlations in networks with different routing but the same product form solution π.

Session WA04 - Networks, Chair: Zazanis in CZ 11

Functional diffusion approximation of the queue with an infinity of servers

L. Decreusefond, ENST Paris, France, [email protected], P. Moyal, Ceremade, Université Paris Dauphine, France, [email protected]

We present the diffusion approximation of the measure-valued process ν representing the remaining pro- cessing times of the customers of a queue with an infinity of servers M/GI/∞. To that end we first deter- mine the mean field (ν¯ ∗) of a sequence {ν¯ n} after a Law of Large Numbers scaling, which is a deterministic continuous√ measure-valued function. Then we prove the weak convergence of the sequence of processes  n (ν¯ n −ν ¯ ∗) to a local martingale, valued in the space S0 of tempered distribution. To prove this, we show (i) the tightness of the sequence in the space of RCLL measure valued processes, (ii) the uniqueness of the subsequential limit by solving an integral equation in S0, the integrated transport equation. We finally apply these results to give fluid and diffusion approximations of the real-valued processes counting respectively the congestion and the workload of the system.

133 11 – Abstracts

Diffusion approximations for reinsurance problems

Michael Zazanis, Athens University of Economics and Business, Athens, Greece, [email protected] A.N. Yannacopoulos, University of the Aegean, Greece, [email protected] V. Papanicolaou, National Technical University of Athens, Athens, Greece, [email protected] E.N. Frangos, Athens University of Economics and Business, Athens, Greece, [email protected]

We study the problem of the joint risk process undertaken by an insurer and a reinsurer via a diffusion approximation. For this model we obtain a partial differential equation, the solution of which provides the probability that at least one of the two firms will be ruined. Using techniques from the theory of large deviations for diffusions we obtain asymptotics for this ruin probability. These are compared to exact results obtained for special cases of the parameters by relating them to the behavior of Brownian motion with drift in Rd . Finally, the problem of optimal reinsurance in this context is also discussed.

Active-Idleness and the Multiclass Queueing Networks Optimization using Infinites- imal Perturbation Analysis

Nuno Órfão, Polytechnical Institute of Leiria, Portugal, [email protected] Carlos Bispo, Institute for Systems and Robotics, Instituto Superior Técnico, Portugal, [email protected]

We address the issue of stochastic multiple class queueing networks optimization using Infinitesimal Per- turbation Analysis (IPA). Previous research introduced the concept of Active-Idleness (which consists on forcing a server to stay idle even in the presence of customers waiting to be served) and demonstrated that is possible to stabilize unstable multiclass queueing networks by slightly changing the scheduling policy. A modified version of the Active-Idleness supervisory controller consisting on the control of relevant end- of-blocking time instants was implemented and lower levels of average work-in-process inventory were achived. A simulation-based optimization algorithm was formulated for the class-fraction queueing net- work control variables, while the IPA is the methodology used to estimate the gradient of the objective function with respect to it. Discrete versions of the derivative estimation equations and of the perturbations propagation rules are provided. The optimization algorithm efficiency is evaluated with several numeri- cal experiments performed with classical multiclass queueing networks (Lu & Kumar, Dai and Seidman models).

On Stochastic Network Calculus

Xiaoyue Jiang, Louisiana State University, USA, [email protected]

This research is to develop a new form of stochastic network calculus for queuing system performance guarantees by making use of the Legendre transform at the sample-path level. A symmetrical formulation is resulted in through separate modeling of traffic flows and service capacities. Performance measures at single nodes and the service capacities of tandem systems are explicitly bounded at the sample-path level which consequently leads to bounds of the corresponding tail probabilities. Comparison and contrast to ex- isting (deterministic and stochastic) network calculus formulations and to the theory of large deviations are performed. The applicability of the proposed approach is demonstrated by recovering the “bottleneck" phe- nomenon studied in large deviations under more general settings and by the improvement of scaling factors known to network calculus. Finally, perspectives on future research directions for tightening performance bounds are provided.

134 WA05

Session WA05 - Internet, Chair: Fricker in CZ 12

Transient analysis and asymptotic of the mean of the first hitting time for a TCP window size process

Andreas H. Löpker, EURANDOM, P.O. Box 513 - 5600 MB Eindhoven, The Netherlands. Email address: [email protected] Johan S. H. van Leeuwaarden, Eindhoven University of Technology and EURANDOM, P.O. Box 513 - 5600 MB Eindhoven, The Netherlands. Email address: [email protected]

The window size in TCP can be modeled as a piecewise deterministic Markov process that increases lin- early in time and experiences downward jumps at Poisson times. We present a transient analysis of this process, the main result being a formula for the Laplace transform of the transient moments. Explicit re- sults for the integer and fractional moments are given, as well as an explicit characterization of the speed of convergence to steady-state. We further present results on the first hitting time of the process, including power series for the mean and the Laplace transform and an analysis of the asymptotic behavior. Analo- gous results are given for the recovery time, that is the time the process needs to recover from a downward jump. Using these results the asymptotic behavior of the maximum process is investigated.

Multiclass G/M/1 queueing system with self-similar input and non-preemptive pri- ority

M. Çaglar˜ , Koç University, Turkey, [email protected] M. Iftikhar, The University of Sydney, Australia, [email protected] B. Landfeldt, The University of Sydney, Australia, [email protected] T. Singh, The University of Sydney, Australia

The Internet domains are tied together by service level agreements which are based on various QoS pa- rameters such as delay, jitter, packet-loss rate, throughput and availability. To offer tighter and more com- prehensive service level agreements, accurate modeling of IP traffic and its queuing behavior over the entire network domain is necessary. We present a novel analytical model of a single router which takes into account multiple classes of self-similar traffic based on G/M/1 queueing system with non-preemptive priority. Our long-range dependent traffic model is generated by infinitely many sources governed by a Poisson random measure. We derive exact expressions for the transition probabilities of the embedded Markov chain of G/M/1 by first deriving the interarrival distribution of the incoming traffic. Closed form expressions for the expected waiting time of multiple classes have been derived. The queuing system is evaluated numerically for a typical router to show the effect of the Hurst parameter on several performance measures. Such work forms a basis for modeling the behavior of self-similar traffic accurately through heterogenous network domains, eventually leading to the formation of realistic service level agreements.

Sampling Internet traffic

N. Antunes, Universidade do Algarve, Faculdade de Ciências e Tecnologia Campus de Gambelas, 8005-139 Faro, Portugal, [email protected] Y. Chabchoub, INRIA, Domaine de Voluceau, Rocquencourt BP.105, 78153 Le Chesnay Cedex, France, [email protected] C. Fricker, INRIA, Domaine de Voluceau, Rocquencourt BP.105, 78153 Le Chesnay Cedex, France, [email protected] F. Guillemin, FT R&D, DAC Lannion, Technopole Anticipa, 2 Avenue Pierre Marzin, 22300 Lannion, France, [email protected] Ph. Robert, INRIA, Domaine de Voluceau, Rocquencourt BP.105, 78153 Le Chesnay Cedex, France, [email protected]

135 11 – Abstracts

The aim of the study is to infer traffic characteristics from a partial knowledge of the traffic, a sampled traffic. The target characteristic is the distribution of the volume of the flows. Such a problem is motivated by pricing issues and detection of attacks. The main result is that statistics as the number of flows sampled k times on a given time interval, can be used to estimate the traffic parameters. Expressions involve means of such variables, and burstiness of traffic has no impact on them. Moreover fluctuations of the statictics are given by a central limit theorem. It also explains why deterministic and random samplings are equivalent. The estimates have been tested on a different types of traffic including ADSL traffic and Abilene traces.

Session WA06 - Infinite Multidimensional Markov Chains, Chair: Squil- lante / Taylor in CZ 13

Geometric decay in a QBD process with countable background states with applica- tions to a join-the-shortest-queue model

H. Li, Mount Saint Vincent University, Canada, [email protected] M. Miyazawa, Tokyo University of Science, Japan, [email protected] Y.Q. Zhao, Carleton University, Canada, [email protected]

A geometric tail decay of the stationary distribution has been recently studied for the GI/G/1 type Markov chain with both countable level and background states. However, it can be only applied to the α-positive case. In this paper, we specialize the GI/G/1 type to a quasi-birth-and-death (QBD) process. This not only refines some expressions, but also allows us to extend the study, in terms of the matrix analytic method, to non-α-positive cases. We apply the result to a generalized join-the-shortest-queue model, which only requires elementary computations. The obtained results enable us to discuss when the two queues are balanced in the generalized join-the-shortest-queue model, and establish the geometric tail asymptotics along the direction of the difference between the two queues.

Stability Analysis of a General Class of Markov Chains

D. Gamarnik, MIT, USA, [email protected] M.S. Squillante, IBM Thomas J. Watson Research Center, USA, [email protected]

In this paper we consider a general class of infinite multidimensional Markov chains and derive a stability analysis of these stochastic systems. More specifically, we establish exact conditions for the stability of this general class of Markov chains using both probabilistic arguments and Lyapunov function techniques, which also elucidates important connections between both sets of related stability arguments based on purely probabilistic means and Lyapunov techniques. We provide large-deviations bounds and algorithmic techniques for the numerical computation of stability conditions and stationary distributions in order to obtain approximations within a given level of accuracy guarantee.

Some interesting decay rate behaviour in quasi-birth-and-death processes with infi- nite phase spaces

Allan Motyer, University of Melbourne, Australia, [email protected] Peter Taylor, University of Melbourne, Australia, [email protected]

Recently there has been considerable interest in the decay behaviour of quasi-birth-and-death processes when the phase space is countably-infinite. It has been observed that processes whose transition structure differs only at level zero can have different decay rates as the level approaches infinity. Furthermore

136 WA07 sequences of truncations in which the phase space is restricted to be finite, and then allowed to approach infinity, can be shown to possess limiting decay behaviour that is not the same as that of the infinite system. In this talk, we shall describe one approach to studying such phenomena and present some examples where this interesting behaviour occurs.

Explicit solutions and decay rates for a general class of Markov chains

J.S.H. van Leeuwaarden, EURANDOM, The Netherlands, [email protected] M.S. Squillante, IBM Thomas J. Watson Research Center, USA, [email protected] E.M.M. Winands, Eindhoven University of Technology, The Netherlands, [email protected]

In this paper we consider a general class of infinite multidimensional Markov chains and derive explicit so- lutions for their stationary distribution and tail asymptotics. More specifically, we present an exact matrix- geometric solution for the stationary probability vector. We reformulate the probabilistic interpretations of the fundamental solution matrices in terms of Bernoulli excursions, leading to explicit expressions for these matrix elements in terms of hypergeometric functions. We establish asymptotic results on the powers of one of the fundamental solution matrices, which are used together with the corresponding matrix element expressions to obtain explicit large-deviation decay rates.

Session WA07 - Optimal Stopping 2, Chair: Sonin in CZ 14

Fiducial approach in the invariant problem of optimal choice

Vitali Z. Belenky, Central Economics & Mathematics Inst.,Russian Academy of Sci., Moscow, [email protected]

Fiducial probabilities were introduced in 1920s by R. Fisher, and were defined rigorously in the works of D. Frazer, R. Hora and R. Bueler and used extensively in the works by G. Klimov and A.Kuzmin. Our goal is to demonstrate that fiducial approach is the most fruitful when a decision-maker, who has to choose the best object, faces a situation of complete uncertainty. If the criterion of choice is invariant or equivariant then it is natural to require the strategy of choice to be invariant as well. But if the criterion of choice has no invariant properties then the question "how to state a problem of choice from distribution Fθ ∈ F with unknown parameter ?" is open. We propose to answer this question we propose on the basis of fiducial probabilities. The main results are as follows: 1. The inertia principle for free random values is formulated and proved. 2. The fiducial sequence is introduced, and its properties are discussed. 3. We propose a logically noncontradictory statement for the problem of optimal choice from distribution Fθ ∈ F with unknown parameter under arbitrary criterion function.

Bandwagon Random Walks

J. Quinn, UNC Charlotte, [email protected]

Some new, basic random walks with dependent increments are introduced. Consider two towers, of possi- bly fractional numbers of boxes, labelled with a ”1” and a ”-1.” Balls are placed in the towers sequentially with probabilities corresponding to the relative proportions of the boxes in the towers. If a ball goes in to one of the towers, a possibly fractional portion of a box is added to that tower for the next ball. The random walks are defined naturally with linear scaling. We determine the asymptotics for the walks, showing that they are governed by beta distributions. While the walks themselves are truly random, their asymptotic

137 11 – Abstracts limits are deterministic after any positive initial time period. Possible applications include situations in finance where, after some initial, uncertain period, a drift direction is picked. Some issues connected with optimal stoping are also discussed.

A Generalized Gittins Index for Markov Chain and its Recursive Calculation

Isaac Sonin, Univ. of North Carolina at Charlotte, USA, [email protected]

We discuss the generalization of the classical Gittins Index for a Markov chain and propose a transparent recursive algorithm for its calculation. The foundation for this algorithm is a modified version of the Elimination algorithm proposed earlier by the author to solve the problem of optimal stopping of a Markov chain in discrete time and a finite or countable state space. [1] Sonin, I.M.,1999, The Elimination Algorithm for the Problem of Optimal Stopping, Math. Meth. of Oper. Res., 49, no.1, 111-123. [2] Presman E.L., Sonin I.M., Gittins Type Index Theorem for Randomly Evolving Graphs, pp. 567-588, in: From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, Kabanov, Y.; Lipster, R.; Stoyanov, J. (Eds.), Springer, 2006.

Session WA08 - Flexible Servers, Chair: Down in CZ 15

Tandem Lines with Synergistic Servers

Sigrún Andradóttir, School of Industrial and Systems Engineering, Gerogia Institute of Technology, Atlanta, GA 30332-0205, [email protected] Hayriye Ayhan, School of Industrial and Systems Engineering, Gerogia Institute of Technology, Atlanta, GA 30332-0205, [email protected] Douglas G. Down, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4L7, Canada, [email protected]

Consider a tandem queueing network with N stations, infinite supply of jobs in front of station 1, infinite room for completed jobs after station N, finite buffers between stations, and M flexible servers. We assume that several servers can work together on a single job and that the servers are synergistic. Our objective is to characterize the structure of the optimal dynamic server assignment policy and to investigate how the optimal policy changes as a function of how well the servers work together.

A call center model with upgrades

Mark E. Lewis, Cornell University, Ithaca, NY, USA, [email protected] Douglas S. Down, McMaster University, Hamilton, ON, Canada, [email protected]

We consider a call center with two parallel queues. The first queue is equipped with several flexible servers that can serve either queue. The second is equipped with dedicated servers. Customers that are not currently in service at the first queue receive upgrades to the second queue after an exponential time in queue. We discuss structural results of optimal allocations of the flexible servers that are both congruent and incongruent with results in models without upgrades.

Resource allocation in grid computing

Ger Koole, Vrije Universiteit, The Netherlands, [email protected] Rhonda Righter, The University of California, Berkeley. USA, [email protected]

138 WA09

In grid computing, with many alternative computers available, each with varying extra capacity, and each of which may connect or disconnect from the grid at any time, it may make sense to send the same task to more than one computer. The application can then use the output of whichever computer finishes the task first. Thus, the important issue of the dynamic assignment of tasks to individual computers is complicated in grid computing by the option of assigning multiple copies of the same task to different computers. We show that when tasks have identical size and computer processing times are NWU, maximizing task replication stochastically maximizes the number of task completions by any time. That is, it is better to do the same task on as many computers as possible, rather than assigning different tasks to individual computers. Computers may be heterogeneous and their speeds may vary randomly, as is the case in grid computing environments. We also show that maximal task replication, along with a cµ rule, stochastically maximizes the successful task completion process when task processing times are exponential and depend on both the task and computer, and tasks have different probabilities of completing successfully.

Linear Programming Based Affinity Scheduling for Heterogeneous Computing Sys- tems

Douglas G. Down, McMaster University, Canada, [email protected] Issam Al-Azzoni, McMaster University, Canada, [email protected]

Resource management systems (RMS) are an important component in heterogeneous computing (HC) sys- tems. One of the jobs of an RMS is the mapping of arriving tasks onto the machines of the HC system. The HC system is modelled as a system of parallel (heterogeneous) queues. Many different mapping heuris- tics have been proposed in recent years. However, most of these heuristics suffer from several limitations. One of these limitations is the performance degradation that results from using outdated global information about the status of all machines in the HC system. Motivated by recent work in the area of queues with flexible servers, we propose a heuristic which addresses this limitation by only requiring partial information in making the mapping decisions and discuss its performance.

Session WA09 - Inventory 3, Chair: Van Houtum in CZ 16

Approximate evaluation of order fill rates for an inventory system of service tools

I.M.H.Vliegen, Technische Universiteit Eindhoven, The Netherlands, [email protected] G.J. van Houtum, Technische Universiteit Eindhoven, The Netherlands, [email protected]

In this talk, we deal with the analysis of a single-location, multi-item inventory model for service tools, in which coupled demands and coupled returns occur. We distinguish multiple Poisson demand streams. Per stream there is a given set of tools that is requested per demand. We are interested in the order fill rates, i.e., the percentage of demands for which all requested tools are delivered from stock. Requested tools that are not on stock are delivered via an emergency channel. For the warehouse under consideration, they may be considered as lost sales. Delivered tools are returned to the warehouse after a deterministic return time, that is equal for all tools. We develop three approximate models for the order fill rates, which are all based on Markovian models. One approximate model has appeared to give an underestimation in all computational tests, while the second approximate model has led to an overestimation in all instances tested. The last approximate model combines the other two. This approximate model is very accurate and can be computed efficiently. Hence, it is appropriate for usage in multi-item optimization algorithms.

Exact and Approximate Results for the Kit Management Problem

R. Güllü, Bogaziçi University, Industrial Engineering Department, refi[email protected]

139 11 – Abstracts

M. Köksalan, Middle East Technical University, Industrial Engineering Department, [email protected]

In this talk we consider a kit planning problem where demand occurrences are not for individual items, but for kits (a group of items). Each kit contains an arbitrary number of items. Kit demands occur according to a Poisson process. Whenever a kit demand occurs, only one item from the kit is used and the rest is returned. The item that will be used from the kit is not known in advance and the whole kit has to stay at the demand site for the whole duration. The used item is replenished through a stochastic supply system, with possible capacity limitation. We present exact results for the kit availability probability, and provide approximate procedures for various performance measures.

Valuation of downstream liquefied natural gas storage

Alan Scheller-Wolf, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA, [email protected] Xiaofeng (Mulan) Wang, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA, [email protected] Sunder Kekre, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA, [email protected] Nicola Secomandi, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA, [email protected]

The U.S. Energy Information Administration forecasts that local natural-gas production will be unable to meet demand in the U.S. and most other developed countries, and expects liquefied natural gas (LNG) imports to play a major role in bridging this demand-supply gap. We present a practical real-option model for the valuation of LNG downstream storage. Our approach is based on a dynamic programming model to determine the optimal release policy of the inventory held at the re-gasification terminal into the wholesale natural-gas market. The model uses a closed queueing network model to represent upstream LNG produc- tion and shipping to the downstream re-gasification facility, and a reduced-form model of the evolution of the natural-gas spot price. We show that the optimal LNG inventory-management policy at the downstream facility has a basestock-target structure that depends on the natural-gas spot. The structure is nontrivial: When it is optimal to sell it is not necessarily optimal to sell down to zero. We also quantify the value of the real option to store LNG, and the relative benefits of this option for different parties involved in an LNG chain. We find that the option is equally valuable for LNG merchants and integrated producers, but that its relative benefit is significantly higher for the former than the latter parties.

Analysis of an inventory model with lateral transshipment

A.A. Kranenburg, Centre for Quantitative Methods CQM B.V., The Netherlands, [email protected] G.J. van Houtum, Eindhoven University of Technology, The Netherlands, [email protected]

In practice, spare parts supply chains often consist of multiple local warehouses and one or a few central warehouses. Local warehouses, located close to customers, are used to ensure prompt replenishment of spare parts to customers. Stock replenishment for the local warehouses and emergency delivery to the customers in case local warehouses are out of stock, is done by the central warehouse(s). Since emergency delivery from a central warehouse to a customer can cost a lot of time, often an additional option exists: lateral transshipment. In case a local warehouse is out of stock at the moment of a customer’s request, it is first tried to obtain the required spare part from a neighboring local warehouse, as this will cost less time than an emergency delivery from the central warehouse. If demand at each local warehouse follows a Poisson process and normal replenishment time from the central warehouse to the local warehouses is exponentially distributed, an exact analysis of the situation with lateral transshipment can be done using a Markov Process description. However, this analysis can be

140 WB01 done for situations with a limited number of local warehouses only, where in practice the number of local warehouses typically is large. We present an approximate analysis based on the M/G/c/c-queueing model, and show that our results are close to the exact analysis.

Wednesday 10:30am - 12:00pm

Session WB01 - Last Minute Talks, Chair: Boxma in CZ 4

New Algorithms for Laplace Transform Inversion

Peter den Iseger, Cardano, The Netherlands, [email protected]

The need for advanced numerical techniques in computational probability and finance is rapidly increasing. Many fast and efficient numerical transform inversion algorithms which can be employed to solve many of the problems arising in computational probability and finance have already been published in academic lit- erature. However flexible, fast and precise, these algorithms also have many limitations: they cannot tackle important problems such as iterative algorithms (for instance recursion, widely employed for dynamic pro- gramming style problems), Levy process based modeling, or problems where the inverse transform as well as the transform itself needs to be computed (a good example is the waiting time distribution in a G/G/1 queue). These inversion algorithms perform the inversion in a number of points at a time. On the other hand, the numerical inversion method we are about to introduce is a piecewise inversion: we employ a Gaussian quadrature rule in order to compute the coefficients of the Legendre expansions of the desired function on all intervals [k, k + 1). The two key differences, which enable the new numerical technique to tackle a much wider class of problems than its forebears, can be summarized as follows:

• The original function f (obtained after the numerical inversion) can be evaluated in arbitrary points xk (instead of only in points k, k = 0, 1,...).

• The new method works both ways: if some numerical values of the original function f are given, then the algorithm generates its Laplace transform L f (thus no closed-form analytical expression is needed!), and vice versa: given some numerical values of the Laplace transform, the original function f can be obtained in arbitrary points. These properties enable one to use the inversion algorithm iteratively, hence solving problems such as pricing guaranteed return rate products, fast convolution transforms, Ladder algorithms and the waiting time distribution in a G/G/1 queue. High dimensional versions of the algorithms are very efficient since they exploit a embedded Kronecker product structure and use efficient high dimensional versions of the fast Fourier transform. The precision and robustness of the algorithms is illustrated by a large number of numerical examples.

Session WB02 - Self-interacting Stochastic Processes, Chair: Van der Hofstad in CZ 5

An expansion for self-interacting random walks

M. Holmes, EURANDOM, [email protected]

I will briefly describe joint work with Remco van der Hofstad where we derive an expansion for self- interacting random walks. We will discuss how this expansion can be used to prove monotonicity of the drift for “excited random walk”, as a function of the excitement parameter, when the dimension is sufficiently high.

141 11 – Abstracts

Parabolic Anderson model: Localisation of mass in random media

W. König, Universität Leipzig , Germany, [email protected] P. Mörters, University of Bath, United Kingdom, [email protected] N. Sidorova, University of Bath, United Kingdom, [email protected]

We study the parabolic Anderson problem, i.e., the heat equation ∂t u = 1u + ξu on the d-dimentional d integer lattice with independent identically distributed random potential {ξ(z): z ∈ Z } and localised initial condition u(0, x) = 10(x). Our interest is in the long-term behaviour of the random total mass of the unique non-negative solution, and we prove the complete localisation of mass for potentials with polynomial tails: We find a random process {Zt : t ≥ 0} such that at time t the mass of the system is completely localised P in Zt in the sense that limt↑∞ u(t, Zt )/ z∈Z d u(t, z) = 1, in probability. We also identify the asymptotic distribution of Zt .

Excited to the center

Gady Kozma, [email protected]

Place a cookie on every vertex on Z d and perform a random walk that, upon entering a vertex with a cookie on it, eats the cookie and then does a single step with a constant drift towards the center 0. When it returns to the same vertex and there is no cookie there any more, it does simple random walk. We show that this model is recurrent in all dimensions.

Sampling for the maximum of Brownian motion

Johan S.H. van Leeuwaarden, Eindhoven University of Technology, The Netherlands, [email protected]

Say we observe a standard one-dimensional Brownian motion with negative drift at time points t0 = 0, t1, t2,... only, where the times in between two consecutive points are i.i.d. distributed according to − some r.v. T with ET = ω 1. Here, ω represents the mean number of observations per unit of time, or the sampling frequency. By increasing the sampling frequency the sampled version will converge to the Brownian motion. In this talk we investigate, depending on the choice of T , the speed of convergence as a function of the sampling frequency. We consider three types of sampling: (i) equidistant sampling, where T = ω−1, (ii) random sampling, where T follows an exponential distribution, and (iii) Erlangian sampling, where T follows an Erlang distribution. For comparison, we consider the difference between the all-time maxima of the Brownian motion and its sampled version. This difference tends to zero as ω → ∞. All three sampling methods lead to a random walk of which the distribution of the maximum allows for an explicit solution.

Session WB03 - Stochastic Programming and Optimization 2, Chair: Veatch in CZ 10

Efficient Allocation Under Incomplete Information

Apostolos N. Burnetas, University of Athens, Greece, [email protected]

Consider an allocation problem where one or more resources must be distributed to several activities so that the total benefit is maximized. The benefit from each activity is characterized by a random coefficient

142 WB04 with unknown distribution. We derive an adaptive policy that combines efficiently estimation of the alloca- tion benefits with expected profit maximization. The policy combines ideas from duality in optimization, stochastic approximation and multi-armed bandit theory.

A problem of optimal sequential pricing

D.J. Hodge, University of Cambridge, United Kingdom, [email protected]

We present an investigation into the optimal pricing regime for selling a set of n perishable goods via a sequence of daily prices. The model is one which arises naturally online when purchasing items from websites, where customers see today’s price and choose to either buy today, or shop around and check back tomorrow hoping for a lower price. Under general assumptions we form the stochastic dynamic programming equations and describe how to solve them. We then take a highly plausible specific example and prove some results concerning the monotonicity, or not, of prices over time. Combining this with some results concerning the character of the optimal value function, we can then quickly find the optimal pricing policy without lengthy search. The application of this problem to the Internet is particularly appropriate. A website administrator is now able, via site statistics, to determine how many customers have requested to see the price and how many of those purchased or returned later.

Cutting speed adjustment and tool magazine capacity allocation using stochastic dynamic programming

B.F. Lamond, Laval University, Canada, [email protected]

We present a stochastic dynamic programming model for the optimal operation of a flexible machine tool, to find a sequence of cutting speeds for the successive tools used in a cutting operation, in order to min- imize expected makespan. We assume the tool life is random and the expected tool life decreases with cutting speed, and each tool requires a setup time when manually installed. Using a numerical method for computations, we compare the optimal dynamic policies with well-known static policies and we examine their impact on tool magazine capacity allocation among several part types and on expected makespan. Our results explain why dynamic adjustment of cutting speed is better with random tool life.

Approximation, structure and performance for average cost queueing network con- trol

Michael Veatch, Gordon College, Wenham, Massachusetts, [email protected]

Differential cost approximations yield tractable optimality equations, solved as an LP, for moderate size queueing networks. For small examples, analysis of the structure of this LP allows characterization of the average cost bound and the associated policy. This analysis and interpretation of the dual assist in choosing approximations with good performance. Numerical results are given for larger networks.

Session WB04 - Fluid Limits and Mean Fields, Chair: D’Auria in CZ 11

Mean field analysis of discrete-time closed queueing networks with geometric ser- vice times.

I.J.B.F. Adan, Eindhoven University of Technology, The Netherlands, [email protected]

143 11 – Abstracts

P. Beekhuizen,EURANDOM, The Netherlands, [email protected] B. D’Auria, Universidad Carlos III de Madrid, Spain, [email protected]

We consider a discrete-time closed queueing network with M servers and N circulating customers. At each time the servers have equal probability p > 0 to serve one customer from their queue if present; the served customers are then routed with equal probability 1/M to any queue of the network. We study the mean field dynamics of this network as the number of servers M → ∞ and N/M → ν = const. We prove that in the thermodynamical limit the random process corresponding to the network dynamic converges to a deterministic dynamical system that is the unique solution for the limiting infinite system of Kolmogorov differential equations. This model is relevant for the analysis of packet switches in ATM and Local Area Networks.

Spontaneous resonances and the onset of collective behavior in queuing networks

A.A. Vladimirov, Institute for Information Transmission, Russia, [email protected] A.N. Rybko, Institute for Information Transmission, Russia, [email protected] S.B. Shlosman, Centre de Physique Theorique, CNRS, Marseille, France, [email protected]

Poisson Hypothesis assumes that a large network goes in a reasonable time to a state where the incoming flows to every node are Poisson ones with almost constant rate. We construct here a closed network, for which this is not the case: almost from every initial state we go to the synchronized state, in which the flow to all the nodes stays periodic in time. The fluid model of this network has a periodic attractor, and we demonstrate that the non-linear Markov process in the mean field mode has a close attractor provided the load is high enough.

Conic Instability of fluid limit models

Bara Kim, Korea University, Korea, [email protected]

We introduce conic instability of fluid limit models and then we show that a stochastic network is unstable in the sense that a representing Markov process for the network is not Harris recurrent, if a corresponding fluid limit model is conically unstable. This work is closely related to Meyn (Annals of Appl. Prob., 1995) and Dai (Annals of Appl. Prob., 1996). Meyn proved that if starting from any (nonzero) initial condition the fluid limit model explodes at a linear rate, then the associated queueing network explodes. Dai introduce weak instability of a fluid limit model which means that every fluid limit starting from zero exits zero after a while, and then he showed that if a fluid model is weakly unstable, then the associated queueing network explodes. The works of Meyn and Dai have pros and cons: Meyn does not have to deal with fluid limits starting from zero, but needs to consider all other fluid limits. Dai has to consider only fluid limits that start from zero. However, fluid limits starting from zero are most difficult to be characterized in some examples. Conic instability introduced in this work is determined by only fluid limits starting from nonzero points in a conic region. We give some examples where the conic instability is suitably applied. The result of this work can be used as one of general tools that show instability of stochastic networks.

Session WB05 - TCP, Chair: Baccelli in CZ 12

Iterated function systems and ergodicity of AIMD networks

C. King, Northeastern University, USA, [email protected] R. Shorten, The Hamilton Institute, NUI Maynooth, Ireland, [email protected]

144 WB05

F. Wirth, University of Bremen, Germany, [email protected] D. Leith, The Hamilton Institute, NUI Maynooth, Ireland, [email protected]

In this talk we consider communication networks that employ drop-tail queueing and Additive-Increase Multiplicative-Decrease (AIMD) congestion control algorithms. A basic assumption in the study of such networks is that the underlying stochastic process is ergodic. Here we present a proof of ergodicity under very general assumptions. Fairness formulae for the average congestion window size, and for throughput are also given. Our starting point is a recently derived algebraic model that captures the essential features of networks in which TCP is deployed. This model may be analyzed using methods from matrix semigroup theory, in particular results on paracontractive sets of matrices. This observation leads to the consideration of the model as an iterated function system with a contractivity property for which several ergodicity results are available. Extensions to nonlinear and adaptive versions of congestion control algorithms will be discussed.

Queueing Theory for TCP

D. Wischik, University College of London, [email protected]

Most traffic in the Internet is controlled by TCP, an algorithm which adjusts the transmission rate of a traffic flow in response to the congestion it perceives in the network. In the core of the Internet, where there are many TCP flows, the aggregate traffic behaves predictably, both at a stochastic level and a fluid level. This predictability is the basis for queueing theory for TCP. I will describe the natural limiting regime which permits us to obtain limit theorems for TCP, and show how the stochastic and fluid parts of the limit mesh together. In fact, there are three different ways they can mesh together, depending on how big the queue’s buffer is – by changing the buffer size we obtain different features of the TCP models of Baccelli, Kelly, and Srikant. I will also describe the sorts of traffic descriptors which are appropriate for capturing the burstiness of TCP traffic. I will finish with practical application of this work, an answer to the question: how big should buffers be in core Internet routers?

A stochastic model of Split TCP

F. Baccelli, INRIA ENS, Paris, franç[email protected] G. Carofiglio, Politecnico di Torino/INRIA ENS, Paris, giovanna.carofi[email protected] S. Foss, Heriot-Watt University, Edinburgh, [email protected]

In wireless networks, TCP is proved to poorly perform by over-reacting to losses due to radio transmission errors that it wrongly attributes to congestion. The “split-connection” approach has been proposed in order to achieve higher throughput in mixed wired-wireless scenarios, where it operates a separation at transport layer between the wired and the wireless part. According to Split TCP, each TCP connection traversing two different media is split into two halves and a relay node is placed in the middle to temporarily store in-transit packets. In the model we developed we aimed at the characterization of TCP throughput on both links as well as of buffer occupancy at the relay node. A representation of the system is given in terms of piecewise- deterministic Markov process and its stability is proved by exploiting some sample pathwise continuity and monotonicity properties. We also analyzed the tail asymptotics for the buffer occupancy in the stationary regime.

145 11 – Abstracts

The Equations of TCP

Francois Baccelli, École normale supérieure, Paris France, [email protected] Ki Baek Kim, Samsung Electronics Co., Suwon, Korea, [email protected] David McDonald, University of Ottawa, Ottawa Ontario, Canada, [email protected]

We present a class of transport equations arising from stochastic models of congestion control. This class contains two cases of loss models as particular cases: the rate-independent case where the packet loss rate is independent of the throughput of the flow and the rate-dependent case where it depends on it. This class of equations covers both the case of persistent (TCP) flows and of non-persistent (HTTP) flows. There is a unique density solving the associated differential equation. This density and its mean value are provided as closed form expressions thereby generalizing the well know "square root formula" for persistent TCP flows to non-persistent HTTP flows.

Session WB06 - Call Centers 3, Chair: Koole in CZ 13

Modeling arrivals

Philippe Chevalier, Université catholique de Louvain, Belgium, [email protected] Jean-Christophe Van den Schrieck, Université catholique de Louvain, Belgium, [email protected]

The peakedness of an arrival flow is an indirect measure of the variability. It can be used to characterize arrivals in a much broader way than more conventional renewal models. In this talk we try to highlight some properties of the peakedness as measured on some real data collected in different settings.

An approximation method for evaluating the performances of a skill-based routing call center model

Oualid Jouini, Ecole Centrale Paris, France, [email protected] Mohamed Salah Aguir, Ecole Superieure de Technologie et d’Informatique, Tunisia, [email protected] Yves Dallery, Ecole Centrale Paris, France,[email protected]

We consider a call center with multiple types of customers and multiple groups of specialized agents. Customers of a given type are only served by an agent of their corresponding group. In addition, there is a flow of customers having “general" questions that can be served by agents of any group. Such a situation can be encountered in various applications such as bank call centers. First, we propose a skill- based routing architecture to operate our system. We develop a model in which we define the rule of assigning “general" customers to specialists. Next, we develop an approximation method to numerically evaluate the performance measures of each type of customers. The performance measures of interest are related to queueing delays. We conduct an extensive simulation study in order to examine the accuracy of the approximation method. The comparison between numerical and simulation results shows that the approximation method is fairly accurate. Finally, we demonstrate how “general" customers are useful to counterbalance the lack of pooling due to only having specialist groups. It appears, indeed, that a small proportion of “general" customers allows to approximately attain the same performances as in a fully pooled structure.

Design and Capacity Planning in Multi-Skill Call Centers

E.L. Örmeci, Koç University, Turkey, [email protected]

146 WB07

N. Çakan, Georgia Institute of Technology, U.S.A., [email protected] Z. Ak¸sin, Koç University, Turkey, [email protected] F. Karaesmen, Koç University, Turkey, [email protected]

Call centers typically receive different types of calls, which require different skills from the agents. At one extreme, agents can be trained to answer only one type of calls, which decreases the training costs at the expense of operational inefficiency, while at the other extreme they can be trained to answer all types to achieve high operational efficiency, only with high training costs. Obviously, a wide range of possible training options lies in between the two. Hence, one of the major issues in call centers is to decide on the cross-training level of agents, or equivalently to design the skill-sets available to the agents as well as the staffing levels in each set. We first model the problem to determine the staffing levels for a given skill-set design as a two-stage stochastic program, which is solved by a simulation-based optimization procedure using sample-path gradient estimation. Then we compare the performance of different skill-set designs under different scenarios on the call rates, profits and training costs.

Dynamic call blending in the presence of time-varying arrivals

A. Balint, VU University Amsterdam, The Netherlands, [email protected] S. Bhulai, VU University Amsterdam, The Netherlands, [email protected]

Call blending (dynamically mixing inbound and outbound traffic) is an efficient method in call centers for obtaining high productivity and high service levels simultaneously. For time-homogeneous Poisson arrival processes, a near optimal policy assigns agents to outbound calls if the number of idle agents exceeds a certain threshold. Based on this policy, we investigate how call blending can be implemented dynamically in the presence of time-varying arrival processes. We develop easily implementable methods resulting in a considerable improvement compared to the static near optimal policy with a fixed threshold and traditional forecasting methods.

Session WB07 - Dynamic Control for Service Management, Chair: Hampshire in CZ 14

A Modified Fluid Model for the Optimal Design of a Dynamic Call Center

Bill Massey, Princeton University, [email protected]

A wide range of real time services can be modeled as loss systems. Numerous examples arise in the service industry including: agent staffing for call centers, provisioning bandwidth for private line services, making rooms available for hotel reservations and congestion pricing for parking spaces. In general these services charge arriving customers a price for admission and customers expect a specific quality of service level. Given that arriving customers make their decision to join the system based on the current service price, the manager may use price as a mechanism to control the utilization of the system. A major objective for the manager is then to find a pricing policy that maximizes total revenue while meeting the quality of service targets desired by the customers. For systems with growing demand and service capacity, we provide a dynamic pricing algorithm. A key feature of our solution is congestion pricing. Our solution uses demand forecasts to anticipate future service congestion and set the present price accordingly. The optimal price is a function of the number of customers in the system and a “future congestion tax". We also quantify revenue tradeoffs between service capacity expansion and quality of service changes.

147 11 – Abstracts

Inventory rationing for a system with a lost-sales and a backordering customer class

P. Enders, Carnegie Mellon University, USA, [email protected] I.J.B.F. Adan, Technische Universiteit Eindhoven, The Netherlands, [email protected] G.J. van Houtum, Technische Universiteit Eindhoven, The Netherlands, [email protected]

We consider a single product that is kept on stock in a single stockpoint. A first class of customers has high priority and demands from this class are lost if they are not immediately satisfied from stock. A second class of customers has a lower priority and those demands are backordered if they cannot be satisfied from stock. For this class, one may even decide to backorder demands when the physical stock is low and one rather keeps those last items on stock for possible high priority demands. Demands arrive according to Poisson processes for both customer classes, we assume a deterministic replenishment leadtime, and we distinguish linear inventory holding and penalty costs. We consider the class of critical level policies, under which the inventory position is kept at a fixed basestock level and low priority demands are backordered if the physical stock is at or below a fixed critical level at the moment of arrival. The objective is to minimize average costs. We discuss the evaluation of a critical level policy via a matrix-geometric approach, we derive monotonicity results via Markov reward theory, we develop a method for the optimization within the class of critical level policies, and we study the distance to the real optimal solution.

Dynamic Pricing and Capacity Planning of e-Services under Demand Uncertainty

Cathy Xiu, IBM TJ Watson Research

In IT services markets, the demand for a service offering is generally governed by the price and the quality of service. Customers have different profiles and they respond differently to price and service level offer- ings. The service level that a firm can offer depends upon its capacity and also the demand. Therefore accurate demand estimation requires the incorporation of the inherent interdependence between demand, price, quality of service and capacity. In this paper, we present a dynamic modeling framework to capture such interdependence. We further propose novel solutions for dynamic pricing and capacity planning under demand uncertainty.

Efficient Frontier Dynamic Pricing for On-Demand Music and Movie Services

Robert C. Hampshire, Carnegie Mellon University, [email protected]

The online distribution of music, television programming, and movies is growing. The recent emergence and popularity of iTunes, Real, Napster, eMusic, YahooMusic, and a host of other digital media distribution services is evidence of an expanding marketplace. However, these services face demand fluctuations, uncertain advertising revenue, and high costs for obtaining digital content. Managers of digital media distribution services seek to increase the operational efficiency of their busi- nesses. A strategy for pricing the digital media content under dynamic demand and in the presence of licensing costs are essential for survival. We develop dynamic pricing policies that directly consider de- mand fluctuations and the variability in the amount of content downloaded by customers. We view the service price as a mechanism to hedge demand and purchasing uncertainty. Pricing policies are presented for both the risk neutral and risk sensitive manager. In the risk neutral case, we analytically characterize the optimal service price for the bounded elastic demand case in terms of a competitive service price . Additionally, for the risk neutral case, we provide a tool for managers to determine whether a free advertisement supported service or a user supported service is most appropriate.

148 WB08

Session WB08 - Queueing Models 3, Chair: Brandt in CZ 15

Necessary and Sufficient Conditions for Finite First Moment of the Workload Com- ponents in FIFO Multiserver Queues

A. Scheller-Wolf, Carnegie-Mellon University, USA, [email protected] R.A. Vesilo, Macquarie University, Australia, [email protected]

In this talk, we derive first moment results for all the components of the stationary workload vector in stable FIFO GI/GI/s queues (previous work has dealt only with the smallest component of the vector, the delay) . Our moment conditions incorporate the interaction between service time distribution, traffic intensity (ρ) and the number of servers in the queue. Sufficient conditions so that EWi < ∞, where Wi is the i’th smallest component of the ordered workload vector, depend crucially on the traffic intensity relative to i, specifically on whether i ≤ k + 1 or i > k + 1, where ρ satisfies k < ρ < k + 1.

To be precise, if we denote a generic service time random variable by S, For i ≤ k +1, EWi < ∞ provided that ESβ(i) < ∞, where β(i) = (s − k + 1)/(s − k). β(i) For i > k, EWi < ∞ provided that ES < ∞, where β(i) = (s − i + 1)/(s − i). β Finally for S in the class L1 (i), defined in [2], these conditions are also necessary. References [1] Scheller-Wolf, A., Necessary and Sufficient Conditions for Delay Moments in FIFO Multiserver Queues: Why s Slow Servers are Better than One Fast Server for Heavy-Tailed Systems. Operations Research, 51, pp. 748 - 758, 2003. [2] Scheller-Wolf, A. and Vesilo, R., Structural Interpretation and Derivation of Necessary and Sufficient Conditions for Delay Moments in FIFO Multiserver Queues, Queueing Systems, 54, 221 - 232, 2006.

Comparison on Queueing Systems with Generalized Processor Sharing Scheduling

Sunggon Kim, Department of Information Statistics, Gyeongsang National University Ju Yong Lee, Department of Electrical & Computer Engineering, University of Toronto Wonyoung Park, Department of Electrical Engineering and Computer Science, KAIST

Generalized processor sharing(GPS) service policy is a scheduling algorithm to allocate the bandwidth of a queueing system with multi-class input traffic. The service policy is work-conserving and the backlogged queues share the full bandwidth according to the preassigned ratios. Thus, the service policy guarantees a minimum bandwidth for each class of traffic while it yields the multiplexing gain. In a queueing system with single-class traffic, the performance such as waiting time or the overflow probability to a threshold becomes to be better if the bandwidth(i.e. the service rate) of the system increases. We consider the same problem for GPS systems. We obtain a sufficient and necessary condition under which one queueing system shows performance at least as good as the other for any input process. We also consider the mode general problem. Suppose that some guaranteed bandwidth of a system is given and we want to determine the other guaranteed bandwidths so the this GPS system shows performance at least as good as a given GPS system. We obtain a condition so that we can calculate the other guaranteed bandwidths. In numerical results, we show that all performance do not deteriorate if the guaranteed bandwidths satisfy the proposed condition. We also investigate the sharpness of the sufficient conditions by numerical study. We expect that the proposed condition can be a guideline for changing GPS parameters when QoS requirements change.

Approximations for the M/GI/N + GI Type Call Center

Foad Iravani, University of Toronto, Department of Mechanical and Industrial Engineering, 5 King’s College Rd., Toronto, ON M5S 3G8, Canada, [email protected]

149 11 – Abstracts

Barı¸sBalcıoglu˜ , University of Toronto, Department of Mechanical and Industrial Engineering, 5 King’s College Rd., Toronto, ON M5S 3G8, Canada, [email protected]

In this study, we propose approximations to compute the steady-state performance measures of the M/GI/N+ GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1 + GI queue for which we suggest a practical way to compute the waiting time distributions and their moments. To do this, we make use of the Laplace transform of the workload density function. Our first contribution is within the context of the M/GI/1+ GI queue in obtaining the workload Laplace transform when abandonment-time distributions are of Phase-type. Next, we scale-up the M/GI/1 + GI queue giving rise to our approxima- tions to capture the behavior of the multi-server system. As a practical application area, we further employ the approximations in determining the minimum staffing levels in a call center to meet the target values of certain performance measures, such as serving a proportion of customers within a specified time, and keeping the portion of abandoning customers below set limits.

Waiting times for M/M systems under generalized processor sharing

M. Brandt, Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Germany, [email protected] A. Brandt, Humboldt-Universität zu Berlin, Germany, [email protected]

We consider a system where the arrivals form a Poisson process and the required service times of the requests are exponentially distributed. According to the generalized processor sharing discipline, each request in the system receives a fraction of the capacity of one processor which depends on the actual number of requests in the system. We derive systems of ordinary differential equations for the LST and for the moments of the conditional waiting time of a request with given required service time as well as a stable and fast recursive algorithm for the LST of the second moment of the conditional waiting time, which in particular yields the second moment of the unconditional waiting time. Moreover, asymptotically tight upper bounds for the moments of the conditional waiting time are given. The presented numerical results for the first two moments of the sojourn times in the M/M/m − PS system show that the proposed algorithms work well.

Session WB09 - Combinatorial Structures, Chair: Gamarnik / Ra- manan in CZ 16

Probabilistic analysis of game tree evaluation

R. Neininger, J.W. Goethe University Frankfurt, Germany, [email protected]

In the analysis of game-searching methods a basic problem is to determine the value of the root of a minimax tree with certain given numbers at its leaves (the input). Various models for the input and related evaluation algorithms have been proposed and analyzed. We review some of these models, in particular models with input from the set {0, 1} and probabilistic mod- els such as Pearl’s model and the (random) incremental model. We discuss the complexity of evaluation algorithms under these models: a new tail bound for the complexity of Snir’s randomized evaluation algo- rithm is given improving upon a Gaussian tail bound due to Karp and Zhang. Also a limit law for the root’s value in Pearl’s model is given leading to a conjecture on the asymptotic distribution of the complexity of α − β pruning. The talk is based on joint works with Ali Khan and with Ali Khan and Devroye.

150 WC01

On Temperate Regions in Random Groves

Y. Baryshnikov, Alcatel-Lucent, Bell Labs, New Jersey, [email protected]

Random groves process studied by Petersen and Speyer exhibits a frozen/temperate type of behavior similar to what is seen in random tilings of Aztec diamonds. In this work we justify these simulational observations by finding the exact asymptotics for probabilities of edges using the complex analytic techniques. Joint with Robin Pemantle.

The mixing advantage is less than 2

Kais Hamza, [email protected]

Given n positive random variables X1,..., Xn, with specified expected values of n independent and iden- tically distributed copies of X1,..., Xn respectively, we obtain upper and lower bounds on the expected value of the maximum of n independent copies of X1,..., Xn. We also present related comparison results.

Iterative Algorithm for Maximum Weight Independent Set

S. Sanghavi, Massachusetts Institute of Technology, [email protected] D. Shah, Massachusetts Institute of Technology, [email protected] A. Willsky, Massachusetts Institute of Technology, [email protected]

Given a graph G = (V, E), an independent set I is a subset of V such that no two vertices in I are connected in G. When vertices of G are assigned weights, the weight of I is the sum of weights of vertices in it. The question of finding an independent set with the maximum weight arises in communication networks, distributed databases, parallel computation, etc. In these applications, algorithms are required to be distributed for architectural reasons. The popular max-product algorithm for finding the max. weight independent set (MWIS) is an iterative, simple heuristic. However, a little is known about convergence and correctness of the max-product algorithm in the context of MWIS. In this talk, we will discuss the correctness and convergence of a variant of max-product to find max. wt. independent set for any bipartite graph with arbitrary non-negative weights. More generally, we will characterize the performance of this algorithm for arbitrary graph. We will also discuss some implications of convergence of max-product algorithm on integrality gap of a natural LP relaxation of MWIS.

Wednesday 1:30pm - 3:00pm

Session WC01 - Risk and Insurance, Chair: Kolbe in CZ 4

Optimal Insurance Demand and Investment in a Dynamic Mean-variance Frame- work

E. Biffis, Cass Business School, United Kingdom, enrico.biffi[email protected]

We determine optimal insurance and investment strategies for mean-variance agents exposed to sources of risk only partially correlated with traded financial securities. We allow for constraints on the agents’ strategies, uncertainty in model parameters and jumps in the dynamics of relevant risk factors. If the agent is an insurer, we obtain optimal retention levels for proportional reinsurance arrangements and examine when it is optimal to swap insurance risks for financial risks under regulatory constraints.

151 11 – Abstracts

Long-term survivor mixture models for credit scoring

S.F. Mo, City University of Hong Kong, Hong Kong SAR, [email protected] K.K.W. Yau, City University of Hong Kong, Hong Kong SAR, [email protected]

The long-term survivor mixture model, which is an extension of ordinary survival model, is applied for credit risk management for individual borrowers by allowing risk-free (i.e. not at risk) borrowers. Not only estimate the time-to-default of the borrowers, mixture model also predicts the probability of being at risk. Comparing with traditional statistical models, mixture model shows better, or at least comparable, performance for credit scoring. It provides additional information about the borrowers’ default risk, which assists the lending institutions to better manage credit risk.

Stability and equilibria in incomplete markets

Gordan Žitkovic´, University of Texas at Austin, U.S.A., [email protected]

A study of continuity properties of demand functions of risk-averse agents investing in an incomplete market has been initiated in the work of Larsen and Žitkovic´ (2006), and further developed in Kardaras and Žitkovic´ (2007). In this talk I will present a summary of those results and an extension of the fixed-point theorem of Knaster, Kuratowski and Mazurkiewicz beyond the realm of locally convex spaces. Finally, I will present an outline of a research program where the combination of the two is used to estabilsh existence of financial equilibria in incomplete markets.

Valuation and pricing of reverse mortgages

Andreas Kolbe, HVB-Institute for Mathematical Finance, Munich University of Technology, Germany, [email protected] Rudi Zagst, HVB-Institute for Mathematical Finance, Munich University of Technology, Germany, [email protected]

Reverse mortgages are an innovative loan product for older homeowners. The homeowner receives either a lump sum or a fixed annuity from the mortgage lender. The reverse mortgage contract is terminated when the mortgagor dies or sells the house. At this (random) point of time, all outstanding debt including all accrued interest or, if the outstanding loan amount exceeds the house value, the house sale proceeds are paid back to the mortgage lender. We develop a consistent valuation framework for reverse mortgage contracts based on reduced-form in- tensity models as used in credit risk modelling. Within our modelling framework we explicitly calculate the probability that the total loan amount exceeds the house value at termination of the contract and derive the maximum payment(s) which can be made to the homeowner under certain constraints. We apply our results to data from Germany, where reverse mortgages are not yet available, and discuss implications for the design of reverse mortgages from a lender’s perspective.

Session WC02 - Spatial Models 2, Chair: Foss / Stolyar in CZ 5

Dynamics of pattern formation in multi-hop wireless networks using backoff-based MAC protocols

Mathilde Durvy, LCA3-ISC-IC, Station 14, EPFL, CH-1015 Lausanne, Switzerland, mathilde.durvy@epfl.ch Patrick Thiran, LCA3-ISC-IC, Station 14, EPFL, CH-1015 Lausanne, Switzerland, patrick.thiran@epfl.ch

152 WC02

The design of Medium Access Control (MAC) protocols for multi-hop ad hoc networks is challenging, as each network entity has only a local view of the network. Consequently, efficient, decentralized, MAC protocols are needed. In this talk, we demonstrate that a protocol based on a local backoff mechanism (similar to the popular IEEE 802.11 protocol) can globally organize the transmissions in a network, in patterns of simultaneous non interfering transmissions. We use a Markovian loss network model with local interactions to explain why such MAC protocols can achieve the maximal level of spatial reuse of the wireless medium (and a throughput close to optimum), but also why they can lead to severe unfairness among competing nodes – they can even favor spontaneously some connections to such an extent that the others suffer from virtually complete starvation. Using this model, we analyze the impact of different parameters of the network and protocol on the spatial reuse and fairness.

Optimal throughput allocation and stability in a spatial random-access model

A.L. Stolyar, Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USA, [email protected]

We consider a model of random access (“Slotted-Aloha-type”) communication networks of general topol- ogy. Assuming network links receive exogenous arrivals of packets for transmission, we seek dynamic distributed random access strategies whose goal is to keep all network queues stable. We consider two such strategies, both queue length based, and show their stability, as long as input rates are within the network saturation rate region.

A Random Multiple Access Protocol with Spatial Interactions

C. Bordenave, University of California at Berkeley, USA, [email protected] S. Foss, Heriot-Watt University, Edinburgh, UK, [email protected] V. Shneer EURANDOM, Eindhoven, The Netherlands, [email protected]

We adopt the classical ALOHA scheme to the case when the transmissions in a multiple-access system create noise in a certain neighbourhood of the receiver. We prove that the fluid limits of the system’s workload satisfy a certain differential equation and, via an analysis of this differential equation, obtain conditions sufficient for the system to be stable. We also discuss the behaviour of the fluid limits.

Routing in Sensor Networks by Using Spatial Dynamics

Anthony Ephremides, University of Maryland, [email protected]

The problem of routing in ad hoc wireless networks has been heavily worked on but mainly from the perspective of traditional delay performance criteria. In wireless sensor networks the performance criteria are different. There is a need to conserve energy and a need to perform the mission of the sensor network. We focus on networks that are deployed to detect the presence of targets. Thus the performance criterion is the probability of correct detection and the concurrent achievement of energy efficiency.The entire network must be designed to support these objectives. The determination of optimal routes is prohibitively complex unless the performance criteria can be mapped into link metrics. In our approach we exploit the spatial dynamics of the signals emitted by the target (and measured by the sensors) to achieve such a mapping. We obtain actually a simple link metric as a result that is, however, predicated on a tandem arrangement of sensors. As a result the use of this metric alone for performing routing is unsatisfactory. Heuristic "fixes" can be applied to translate the result to practical schemes. However, a truly two-dimensional approach is also possible and we outline its preliminary formulation. This presentation is based on joint work with several people from Cornell University, notably Lang Tong and his students, or ex-students, Youngchul Sung, Saswat Misra, and Animashree Anandkumar.

153 11 – Abstracts

Session WC03 - Perturbation and Discrete Events, Chair: Heidergott in CZ 10

Law of Large Numbers for products of random matrices in the max-plus semiring

G. Merlet, CEREMADE, Université Paris-Dauphine, FRANCE, [email protected]

We analyze the asymptotic behavior of the sequence of random variables (x(n, x )) defined by x(0, x ) = 0 n∈N 0 x and x(n + 1, x ) = A(n)x(n, x ), where (A(n)) is a stationary and ergodic sequence of random ma- 0 0 0 n∈N trices with entries in the semiring (R ∪ {−∞}, max, +). Such sequences model a large class of discrete event systems, among which timed event graphs, 1-bounded Petri nets, some queuing networks, train, communications or computer networks. The sequence satisfy a strong law of large numbers (LLN) means that the asymptotic output or throughput exists. Thus, there has been many paper about this subject, under various additional hypotheses. Using results by J. Mairesse and T. Boush on the backward scheme for such sequences, we give a set of necessary conditions for the LLN to be satisfied. Thanks to the interpretation in terms of paths on weighted graph, techniques of subadditivity and results on finite state Markov chain, we prove a theorem to have converse results. This theorem implies all previous LLN for such sequences and allows us to prove that one of the necessary condition is sufficient when the matrices are i.i.d. .

Optimal short-term utilization bounds on multiclass networks

N. M. Órfão, Polytechnical Institute of Leiria, Leiria, Portugal, [email protected] C. F. Bispo, Instituto de Sistemas e Robótica, Instituto Superior Técnico, Lisbon, Portugal, [email protected]

For multiclass queuing networks operated with distributed scheduling policies, previous work proposed the utilization of idling policies to address the stability problem. A controller, termed Time-Window Controller, was constructed where each class has a short-term bound on the amount of time it is allowed to occupy its server. Networks previously shown to be unstable were stabilized by an instance of this controller, not by changing policy, but by allowing the original policy to become non work-conserving. Also, some preliminary simulation results showed the controller to have the potential to improve performance of any given distributed scheduling policy, even for stable networks. To evaluate the controller’s ability to improve the performance of a given distributed scheduling policy, it is necessary to develop an efficient methodology to determine the optimal short-term bounds. We will address this problem using Infinitesimal Perturbation Analysis, IPA, to compute gradient estimates from a single simulation run. The expressions for the derivatives and their respective propagation rules will be introduced. The validity of the IPA approach as a method to produce unbiased gradient estimates will be addressed, and some preliminary computational studies will be presented, and their consequences discussed.

Some new results on the performance optimization of queueing systems

Li Xia, IBM China Research Laboratory, [email protected] Wenjun Yin, IBM China Research Laboratory, [email protected] Jin Dong, IBM China Research Laboratory, [email protected]

Concerning a closed Jackson network with M servers and N customers, we propose a new method to op- timize the service rates. The objective is to minimize (or maximize) the customer-average performance, which is different from the traditional time-average performance. As we know, the steady-state distribu- tion of closed Jackson networks has the product-form solution. We use this special property to aggregate

154 WC04 the perturbation realization factors, which are fundamental concepts of the Perturbation Analysis (PA) in queueing systems. With this aggregated perturbation realization factors, we derive the performance differ- ence equation when the service rates are changed. Then, a type of policy iteration algorithms is developed. The complexity of this algorithm (the number of aggregated perturbation realization factors) is linear to the system size, although the system state space will grow exponentially. Thus, our method survives the “dimensionality curse" of this type of problems. Simulation results demonstrate its efficiency. Our results can be viewed as a complementary method for the traditional Perturbation Analysis in queueing systems, which uses the gradient-based algorithms to do optimization.

Sensitivity analysis for Gaussian systems, a mathematical “urban legend” busted

B. Heidergott, Vrije Universieit, The Netherlands, [email protected] F. Vázquez-Abad, The University of Melbourne, Australia, [email protected] W. Volk-Makarewicz, The University of Melbourne, Australia, [email protected]

Infinitesimal Perturbation Analysis (IPA) is a methodology that uses a stochastic derivative to estimate the gradient of the expected value of a performance function, with respect to some parameter of the model. It is unbiased only when the expectation and derivative operations can be interchanged. For almost two decades, many papers on gradient estimation for discrete event driven systems have claimed that IPA is the preferred method, if applicable, and it is broadly accepted today as the method with minimal variance. The Score Function (SF) and Weak Derivative (WD) methods can deal with discontinuities. They build directly the derivative of a probability measure with respect to some parameter of the model. In this talk we will focus on a ubiquitous class of problems where the dynamics of the system follows a Gaussian model. They arise whenever the processes driving uncertainties are described by "cumulative small errors", and are used in Physics, Chemistry, Economics, Finance, Engineering, etc. We will present theoretical results that show that the WD estimator for the derivative with respect to the variance of a normal random variable actually has smaller variance than IPA, when the performance is any polynomial function of the variable, provided that the degree is higher than 1. For an exponential, the parameter space is divided into two distinct regions where either IPA or WD exhibit smaller variance. These results contradict and provide the first rigourous proof that the mathematical "urban legend" is, in fact, unfounded. We will discuss the results for a more complex Gaussian system using a stochastic activity network, where we compare the three methods and we will discuss applications of our methodology to the problem of sensitivity estimation when pricing financial derivatives. The potential impact of our results is the encoding of a robust “one-size-fits-all" kind of algorithm to estimate the gradients of Gaussian systems performances with respect to either the mean or the variance of the error terms. It is robust because WD (like SF) is a measure-valued derivative estimator that is constructed from the derivative of the joint normal distribution, and it is unbiased for a large class of performance measures. To make our method efficient, we have optimised the CPU time for the generation of the random variables required, as well as introduced a particular correlation that may reduce the variance of the final estimators.

Session WC04 - Large Deviations 2, Chair: Ignatiouk-Robert in CZ 11

Large Deviation and Fluctuation of Diffusion Processes with discontinuous drift

T.S.Chiang, Academia Sinica, Taiwan, [email protected] S.J.Sheu,Academia Sinica, Taiwan, [email protected]

155 11 – Abstracts

For the system of d-dim stochastic differential equations

d X  = b(X )dt + dW(t), t∈[0, 1] X  = x0 ∈ Rd where b(.) is smooth except possibly along the hyperplane x1 = 0, we shall present a large deviation theorem for small . If in addition the drift satisfies the stability condition b1(x) < 0 if x1 > 0 and  b1(x) > 0 if x1 < 0, we also present a fluctuation theorem for X (t), i.e., we shall find a deterministic function φ(t) such that (X (t) − φ(t))/ = ξ(t) + R(t) where ξ(t) is an Orstein-Uhlenbeck process and |R(t)|→0 in L1 as →0 for each t.

Large deviations asymptotics of exit time and exit location for a class of Lagrangian systems

Agnessa Kovaleva, Space Research Institute, Russian Academy of Sciences, Russia, [email protected]

We construct explicit asymptotic estimates of exit time and exit location for a class of the Lagrange systems with weak additive noise. The appropriate mathematical basis to analyse the long-term escape processes in weakly perturbed systems is large deviations theory. Large deviations principle introduces a deterministic cost (action) functional that must be minimized by the “most likely” exit path. Derivation of the cost criterion and solution of the associated Hamilton-Jacobi equation are the central points of the theory. We introduce the action functional for the Lagrange systems and derive the Hamilton-Jacobi equation of the associated variational problem. The logarithmic asymptotics of the first exit time is found as the solution of the latter equation. The closed-form solution is constructed as a sum of two terms, related to kinetic and potential energy of the system, respectively. The exit location on the boundary of the reference domain and the terminal state of the unconstrained coordinates at the moment of exit is defined. Applicability of the techniques developed to the risk analysis and control problems is illustrated by examples.

Martin boundary with large deviation technique for a random reflected random walks on a half-space

Irina Ignatiouk-Robert, [email protected]

To identify the Martin boundary for a transient Markov chain having Green’s function G(x, y) one have to identify all possible limits limn→∞ G(x, yn)/G(x0, yn) with yn “tending to infinity”. For homogeneous random walks, these limits are usually obtained from the exact asymptotics of the Green’s function G(x, y). For non-homogeneous random walks, the exact asymptotics of the Green’s function is an extremely difficult this problem. We identify Martin boundary for a reflected random walks on a half-space by using large deviation technique combined with the ratio limit theorem. The full Martin compactification is in general not homeomorphic to the “radial” compactification obtained by Ney and Spitzer for the homogeneous d random walks in Z : convergence of a sequence of points yn to a point of on the Martin boundary does not imply convergence of the sequence yn/|yn| on the unit sphere. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.

Application of Tauberian theorem to the exponential decay of the tail probability of a random variable

K. Nakagawa, Nagaoka University of Technology, Nagaoka, Niigata 940-2188, Japan, [email protected]

156 WC05

We give a sufficient condition for the exponential decay of the tail probability of a non-negative random variable. We consider the Laplace-Stieltjes transform of the probability distribution function of the random variable. In the previous work, we proved that if the singularities of the LS transform are only a finite number of poles, then the tail probability decays exponentially. However, this condition is not easy to verify because the axis of convergence is not a compact set. In this paper, we show a theorem that if the abscissa of convergence of the LS transform is negative finite and the real point on the axis of convergence is a pole of the LS transform, then the tail probability decays exponentially. For the proof of the theorem, we extend and apply so-called a finite form of Ikehara’s complex Tauberian theorem by Graham-Vaaler.

Session WC05 - Bandwidth Sharing Networks, Chair: Borst in CZ 12

Bandwidth-Sharing Networks in Overload

R. Egorova, CWI/Eindhoven University of Technology, The Netherlands, [email protected] S.C. Borst, Eindhoven University of Technology/CWI, The Netherlands, Bell Laboratories, USA, [email protected] B. Zwart, Georgia Institute of Technology, USA, [email protected]

Bandwidth-sharing networks as considered by Massoulié & Roberts provide a natural modeling framework for describing the dynamic flow-level interaction among elastic data transfers. Under mild assumptions, it has been established that a wide family of so-called α-fair bandwidth-sharing strategies achieve stability in such networks provided that no individual link is overloaded. In the present talk we focus on α-fair bandwidth-sharing networks where the load on one or several of the links exceeds the capacity. In order to characterize the overload behavior, we examine the fluid limit, which emerges from a suitably scaled version of the number of flows of the various classes. We prove that the fluid limit is a linear function of time, and derive a fixed-point equation for the corresponding asymptotic growth rates. It is further shown that the fixed-point solution is also a solution to a related strictly concave optimization problem, and hence unique. We use the fixed-point equation to investigate the impact of the traffic intensities and the variability of the flow sizes on the asymptotic growth rates.

Stability of bandwidth sharing networks with monotone service rates

Matthieu Jonckheere, CWI, The Netherlands, [email protected]

We examine the stability of multi-class queueing systems with the special feature that the service rates of the various classes depend on the number of users present of each of the classes. As a result, the vari- ous classes interact in a complex dynamic fashion. Such models arise in several contexts, especially in bandwidth sharing networks, and in wireless networks, as resource sharing algorithms become increas- ingly elaborate, giving rise to scaling efficiencies and complicated interdependencies among traffic classes. We summarize existing methods and results for establishing stability in this context and provide an exact characterization of the stability region under certain monotonicity assumptions. We also discuss weaker notions of stability like rate stability. The results are then illustrated for simple examples of networks with two or three classes of users.

Importance sampling in rate-sharing networks

P. Lieshout, CWI, The Netherlands, [email protected] M. Mandjes, University of Amsterdam, The Netherlands, [email protected]

We consider a single-node network, where classes share capacity in a Discriminatory Processor Sharing fashion. Assuming Poisson arrivals and exponentially distributed service requirements for each class, the

157 11 – Abstracts dynamics of the user population may be described by a Markov process. We focus on the probability that, given the network is in some state at time zero, the network is in some other state at a different time. In particular, we assume that the underlying event is rare, such that this probability is extremely small. As no closed-form expressions are available for this probability, an attractive approach may be to resort to Monte- Carlo simulation. Due to the rarity of the event under consideration, however, Monte-Carlo simulation is infeasible. A natural approach to speed up the simulation is to use Importance Sampling (IS). We present an IS algorithm to accelerate the simulation that is based on large-deviations results. Extensive simulation experiments show that the algorithm performs well.

Flow-level Stability of Data Networks with Non-convex and Time-varying Rate Re- gions

J. Liu, Department of Electrical Engineering, Princeton University, NJ, US, [email protected] A. Proutière, KTH, Radio Communication Systems, Sweden, [email protected] Y. Yi, Department of Electrical Engineering, Princeton University, NJ, US, [email protected] M. Chiang, Department of Electrical Engineering, Princeton University, NJ, US, [email protected] H. V. Poor, Department of Electrical Engineering, Princeton University, NJ, US, [email protected]

We investigate the flow-level stability and performance of utility-maximizing allocations in data networks with non-convex and time-varying rate regions. We consider the Markovian flow-level model as adopted in most of the contemporary work, with stationary and independent data arrivals and exponential file size distribution. In case of fixed and non-convex rate region or time-varying rate region, we provide the stability conditions for a large class of allocations, such as the MaxProjection policy and the α-fair policy, and find out that the stability region depends on the particular allocation. Moreoever, for two-class networks under α-fair allocations, we show that the stability region generally decreases when the allocation becomes fairer. This is in sharp contrast with the substantial existing literature on stability under fixed and convex rate regions, in which the stability region coincides with the rate region for many utility-based resource allocation schemes, independent of the value of the fairness parameter. We further study the impact of this fairness-stability tradeoff on the system performance, e.g. average throughput and mean flow response time, and conduct numerical experiments to illustrate the new stability region and the performance vs. fairness trade-off.

Session WC06 - Biology, Chair: Spencer in CZ 13

Determining Hidden Markov Models Efficacy in gene finding problem

A. Kazemnejad , Tarbiat Modares University, Tehran, Iran, [email protected] E. Hajizadeh , Tarbiat Modares University, Tehran, Iran K. Mirjafari , Tarbiat Modares University, Tehran, Iran

One class statistical models is Hidden Markov Models (HMM) which are extensively used to analyze several problems such as speech recognition and finding genes which are implicated in causing cancer. Now a day most of methods for prediction of genes are based on these models. In this paper we show how weighting affects on Generalized HMM efficacy in gene finding problems in eukaryotes. For this aim we choose a simple structure of HMM and entered Open Reading Frames (ORF) information with a weight matrix to model and the appropriate values of weights were estimated based on two criteria of sensitivity and specificity in samples of DNA sequence. Results showed entering weights to model have caused significant effect on efficacy and is comparable to existing gene finding methods.

158 WC07

Count or control? Optimal monitoring for invasive species management

M.A. McCarthy, University of Melbourne, Australia, [email protected] A. Moore, University of Melbourne, Australia, [email protected] P.G. Taylor, University of Melbourne, Australia, [email protected]

There exists an extensive body of research on mathematical models for the optimal control of biological populations. However, in order to apply these models to real world situations monitoring is required (to pa- rameterise models and track population size) and despite being far from trivial, monitoring costs are rarely included when comparing management regimes. We consider the task of controlling an invasive pest popu- lation given a fixed budget which must be split between control and monitoring, where monitoring data are binomial observations of an associated index of the population size. We present a relatively simple model for tracking this index and embed this in a Markov decision model to investigate the trade off between monitoring and control. Preliminary results are somewhat counter-intuitive, and suggest that managers should only monitor when they believe the population is small, that is, in a desirable state. Preliminary results have also demonstrated that a seemingly sensible objective function can lead to somewhat counter- intuitive results, suggesting that it is difficult to intuitively predict optimal monitoring-control regimes. This highlights the need for mathematical rigour in designing management regimes.

Using stochastic epidemic models to determine if emerging avian influenza can es- cape containment

S.E.F. Spencer, University of Nottingham, UK, [email protected] P.D. O’Neill, University of Nottingham, UK, philip.o’[email protected]

Much research effort is currently focused on determining the effectiveness of control measures proposed to halt the spread in humans of an emerging strain of avian influenza, such as the H5N1 subtype. This work uses a multi-type Branching Process of households to assess the probability that an outbreak will escape containment; even if the control measures are thought to be effective. The effect of different infectious period distributional assumptions on the containment probability can also be explored using stochastic orderings.

Session WC07 - Games 2, Chair: Weber in CZ 14

Symmetric Rendezvous Games

J. Fan, Cambridge University, UK, [email protected]

Two people are placed randomly on two distinct places(nodes) of a complete graph Kn or a circle Cn. At each step, they can move to any other place or stay put and are assumed to follow an identical random strategy. The aim is to rendezvous in a minimum time of steps. This is called a rendezvous problem and is like the case that we get separated in a park or a supermarket. It is easy to see that the best strategy when there are only two locations is to visit each one with equal probability at every step. However, the case of 3 locations becomes difficult and is not solved until recently but by a strategy stated in 1990. It is also conjectured that the optimal strategy is not changed if the two players will overlook each other even when they are at the same place. This talk will discuss those various strategies on the rendezvous problem.

On two-stopping and stopping game problems

K. Szajowski, Institute of Mathematics and Computer Science Wrocław University of Technology

159 11 – Abstracts

Wybrzeze˙ Wyspianskiego´ 27, PL-50-370 Wrocław, Poland, [email protected]

The mathematical modeling of economic systems in stochastic environment leads to various mathematical optimization problems. If the decision problem consists in choice of intervention moment one can for- mulate the model of such case as the optimal stopping problem (see e.g. papers by Batabyal and Beladi, Clarke and Reed). If it is allowed to react more than once the approach depends on the number of decision makers and their aims. If there is one decision maker and two reactions we have the optimal two stopping problem. When there are two decision maker and their have prescribe aims we usually treat the problem as the stopping game. In the game approach the communication problem between player should be taken into account. It leads to various concepts of solutions. In this paper some double stopping problems and the two person stopping game models will be presented. The presentation is based on the results by Karpowicz and Szajowski.

Symmetric rendezvous search

R.R. Weber, University of Cambridge, UK, [email protected]

In the symmetric rendezvous search game played on K3 (the completely connected graph on 3 vertices) two players are initially placed at two distinct vertices (called locations). At each step a player can either stay where he is or move to a different location. The players wish to minimize the expected number of steps until they meet. Rendezvous search games of this type were first proposed by Steve Alpern in 1976. They are simple to describe, and model circumstances that most of us have met in real life. However, their solution has proved to be notoriously difficult. The symmetric rendezvous search game on K3 is the first interesting game of this type to have been solved. We discuss its solution, as well as several new tools for addressing such problems. One of these is semidefinite programming (which has been called “linear programming for the 21st century").

Session WC08 - Queueing Models 4, Chair: Vlasiou in CZ 15

Analysis of a queueing system with synchronized services and setup times.

A. Economou, University of Athens, Department of Mathematics Section of Statistics and Operations Research, Greece, [email protected] S. Kapodistria, University of Athens, Department of Mathematics Section of Statistics and Operations Research, Greece, [email protected]

In recent years there is an emerging interest for queueing systems with synchronized services. Such sys- tems occur both in manufacturing (e.g. assembly lines) and in telecommunications (e.g. webseminars). Nevertheless, the literature on this kind of systems is scarce due to the mathematical intractability. In this talk we will present a basic Markovian model which encompasses the characteristics of synchronized ser- vices and setup times. We obtain several performance descriptors concerning the number of customers, the busy period and the waiting time distributions. We show that the theory of basic hypergeometric functions (also known as q-calculus) can be used to simplify some computations and to analyze the limiting behavior of the performance measures.

A Geo/G/1/∞ queueing system with a generalized N-policy and setup-closedown times

P. Moreno, Pablo de Olavide University, Spain, [email protected]

160 WC09

We analyze a discrete-time single-server queueing system whose arrival stream is a Bernoulli process and service times are generally distributed. An extension of the N-policy is considered: the server remains idle till the queue length becomes i ≥ 1 with probability θi . After the idle (vacation) period, the server needs a random amount of startup time before serving. Moreover, after the system is empty in a departure epoch, the service station employs a random time to close down the system, although the shutdown is interrupted and the service is resumed immediately without setup if an arrival takes place during such a period. By using the supplementary variable technique, we obtain the queue and system length distributions as well as some performance measures. We also find out the distributions and mean values of the different periods in a cycle and the waiting time. Finally, a cost model and several numerical results are presented.

Production inventory with service time and vacation to the server

A. Krishnamoorthy, Department of Mathematics, Cochin University of Science and Technology, Kochi 682022, India, [email protected], Viswanath C. Narayanan, Department of Mathematics, Government Engineering College, Sreekrishnapuram, India, [email protected]

A production inventory with positive service time is considered. Whenever the level of the inventory reaches s due to demand (one at a time), a production mechanism is activated. Items are added one at a time. Production continues until inventory level is brought to S. When the inventory level is zero no customer is allowed to join (those who are already in the system are allowed to stay back. We obtain analytical solution to the system state distribution under the assumption of exponential distribution associated with inter-arrival, service and time between two successive additions to the inventory due to production. We then extend the results to the case where arrival of external customers form an MAP; service times are iid PH distributed,; lead times follow what we term as Markovian Lead Time process (MLP). Further no customer is permitted to join the system when the inventory level is zero and when there either no inventory or no customer or both are absent, then the server goes on a multiple vacation. Duration of each vacation is PH distributed with appropriate representation. The system state probability distribution is derived. Several performance measures are computed. Numerical illustrations are provided. An optimization problem is also attempted.

A multi-station queue with dependent interarrival and service times

Maria Vlasiou, Georgia Institute of Technology, USA, [email protected] Ivo Adan, Eindhoven University of Technology, The Netherlands, [email protected] Onno Boxma, Eindhoven University of Technology, The Netherlands, [email protected]

In this talk we discuss a single-server multi-station alternating queue where the interarrival times and the service times are auto- and cross-correlated. We examine two cases. In the first case, interarrival and service times depend on a common discrete time Markov chain. In the second case, we assume that the service times depend on the previous interarrival time through their joint Laplace transform. The waiting time process is directly analyzed by solving a Lindley-type equation by transform methods. Numerical examples are included to demonstrate the effect of the autocorrelation of and the cross-correlation between the inter-arrival and service times.

Session WC09 - Health Care, Chair: Ziya in CZ 16

Optimal outpatient appointment scheduling

G.C. Kaandorp, Erasmus Medical Center Rotterdam, The Netherlands, [email protected] G. Koole, VU University Amsterdam, The Netherlands, [email protected]

161 11 – Abstracts

We study optimal outpatient appointment scheduling. A local search procedure is derived that converges to the optimal schedule with a weighted average of expected waiting times of patients, idle time of the doctor and tardiness (lateness) as objective. No-shows are allowed to happen. For certain combinations of param- eters the well-known Bailey-Welch rule is found to be the optimal appointment schedule. A tool in which this algorithm is implemented can be found on the internet: www.math.vu.nl/obp/healthcare/software/ges.

Nurse Staffing and Bed Capacity within Hospitals Units

O. Jennings, [email protected]

We model a hospital unit as a two-node queueing network. Patients arrive exogenously and, once space is available, are assigned beds. The two nodes represent the states between which patients alternate: needy and content. Needy patients require a nurse to restore them to the content state. Content patients either become needy or are discharged from the hospital. We consider the effect of staffing and bed capacity on system dynamics and quality of service.

Managing New Patient Admissions in Multi-site Healthcare Facilities

E. Tekin, Texas A&M University, United states, [email protected]

Many healthcare providers own and operate multiple facilities that are geographically distributed, possibly within a metropolitan region. Due to random patient arrivals, the allocation of workloads to facilities may not be always balanced, resulting in long waiting times. Hence, some new arriving patients would be willing to switch from their primary choice of healthcare facility to avoid long waits. In this research, we investigate strategies to manage new patient admissions among multiple healthcare facilities to maximize long-run average revenues while minimizing the expected waiting times.

Identifying the Critical Patient: Finding the Balance between Undertriage and Over- triage

S. Ziya, University of North Carolina, Chapel Hill, N.C, U.S.A., [email protected] N.T. Argon, University of North Carolina, Chapel Hill, N.C, U.S.A., [email protected]

Triage is the process of classifying patients into different priority groups according to their injuries. Mis- takes during triage are common and they fall under one of two categories: undertriage, which is assigning a patient a lower priority than what the patient’s condition actually requires, and overtriage, which is assign- ing a higher priority than what the patient’s actual condition calls for. In mass-casualty incidents, a certain level of undertriage and overtriage is unavoidable. However, the triage officers can choose to be either lenient towards undertriage or overtriage with their decisions. In this talk, we use simple queueing models to develop a better understanding of how the decisions need to be made depending on conditions like the patient load and the characteristics of the patients’ injuries. We mainly consider two different scenarios: (i) a war-type environment where there is a constant flow of patients coming in and (ii) a single incident isolated in time where there is some finite number of patients in need of treatment. We provide several insights on the “optimal” triage decisions. For example, we find that the effect of patient load on the triage decisions depends on the injury characteristics of the patients. In particular, we find that when the patients’ injuries are not deadly, with the increasing patient load, the triage officers should be more lenient towards overtriage.

162 12 Author Index

A Baron, Opher, MC09 Budhiraja, A., MC04, TB07, Baryshnikov, Y., WB09 TB07 Aalto, S., MA05 Basrak, Bojan, TA07 Buke, Burak, TA04 Adan, I.J.B.F., MB08, WB04, Bassamboo, A., MA04, TA03, Burnetas, Apostolos N., WB03 WB07 TA06 Buyuk, Buket, TD09 Ak¸sin,Z., TA06, WB06 Bäuerle, N., TD01 Buyukkaramikli, N.C., MA09 Al-Azzoni, Issam, WA08 Bayati, M., TA09 Aldous, David, TD02 Bean, Nigel, TC06 Al-Ibrahim, Assil, TC07 Beekhuizen, P., TC05, WB04 C Alp, O., MA09 Bekker, R., TA05 Al-Seedy, R.O., MC08 Belenky, Vitali Z., WA07 Caglar,˜ M., WA05 Amir, Madjid, MB07 Benichou, Olivier, MB09 Cai, Ning, TA01 Ammar, S.I., MC08 Bensoussan, A., MA01 Cakan, N., WB06 Andradóttir, Sigrún, WA08 Berestycki, J., TB02, TB02 Camia, Federico, MB02 Andrews, D.M., TA04 Berestycki, N., TB02, TB02 Carofiglio, G., WB05 Ano, K., TD07 Berman, Oded, MC09 Cerný, R., TA02 Antunes, N., TB05, TD05, WA05 Bertsimas, D., TA04 Chabchoub, Y., WA05 Areepong, Y., TA07 Bhattacharya, Debasis, TC01 Chaintreau, A., TB04 Argon, N.T., WC09 Bhulai, S., MB05, WB06 Chen, Nan, TA01 Armony, Mor, MB06, MB06 Biermé, H., MA02, MA02 Chen, Zhiyong, MA03 Artalejo, J.R., MA08 Biffis, E., WC01 Chevalier, Philippe, WB06 Asgari, N., TD09 Bijvank, M., MC09 Chiang, M., MC05, WC05 Ata, Baris, MA04 Bispo, C.F., MA07, WA04, Chiang, T.S., WC04 Atar, R., MC04, TB07 WC03 Chick, S.E., MB03 Avrachenkov, K., TC09, TC09 Blanc, Hans, TC08 Choi, Bong Dae, MB05, MB05, Avram, Florin, MC01, MC01 Blanchet, Jose, MA03, MB03, MB05, TC05, TC05, TC05 Awad, H.P., WA03 TA03, TD03, TD03 Choo, Taek Sik, MA08 Ayesta, U., MA05 Blaszczyszyn, B., TA02, TD02 Cil, E.B., TC07 Ayhan, Hayriye, WA08 Blatter, A., TD01 Coakley, Kevin, TD06 Bogachev, L., TB02 Collamore, J.F., TB01 Bordenave, C., TB05, TD02, Cont, R., MA01, TA01 B WC02 Courcoubetis, Costas, TA05 Borst, S.C., WC05 Cudina, M., MC07 Baccelli, F., TA02, WB05, WB05 Botev, Z.I., TC03, TC03 Bae, Y.H., MB05 Boucherie, R.J., MA09, TB08 Baek, Sangkyu, TC05 Boxma, O.J., MA05 D Bai, X.X., MC09, TC01 Bramson, M., MC05 Balcıoglu,˜ Baris, WB08 Brandt, A., WB08 Dabrowski, A., TD04 Balint, A., MB02, WB06 Brandt, M., WB08 Daduna, H., WA03 Ballotta, Laura, TB01 Brun, O., TD08 Dai, Jim, MA05, MB06 Barkai, E., TC02 Dallery, Yves, WB06

163 12 – Author Index

Da Silva, Ana, TC06 Fralix, B.H., MC08 Heideveld, S.A., MA09 D’Auria, B., WB04 Frangos, E.N., WA04 Helber, S., TA06 De¸bicki, K., MC01 Fricker, C., TB05, WA05 Helmes, K., TD07 De Boer, P.T., TA03 Funken, S., TA02 Henderson, Shane G., MA06, Decreusefond, L., WA04 MB03 De Graaf, W.F., TB08 Henken, K., TA06 Degrimenci, I.T., MB06 G Heyde, C.C., MA01 DeGuest, R., MA01 Hodge, D.J., WB03 Gamarnik, D., MB04, TA04, De Kok, Ton, TB09 Holmes, M., WB02 WA06 Denardo, E.V., MA07 Hooghiemstra, Gerard, WA02 Ganesh, A.J., TD02 Den Iseger, Peter, WB01 Hordijk, A., TB03, TB03 Gans, N., MB03 Dieker, A.B., MB08 Hosseinkeh, Vahideh, TA09 Garcia, J.M., TD08 Dimitrova, Dimitrina, TB01 Huang, P., TB09 Gautam, N., TB08 Ding, Hongwei, MA09, MC03 Huang, Wei, TB01 Ghosh, A.P., MC04 Ding, L., TD06 Hult, H., TD01, TD03 Giardina, C., MC02 Dolgin, A., TC03 Hwang, Eunju, TC05, TC05 Gibbens, R.J., TA05 Donato, D., TC09 Hwang, Gang Uk, TD05 Giesecke, Kay, TA01 Dong, Jin, MA09, MC03, MC09, Glasserman, Paul, MA03, MA03, TC01, WC03 TA01 Doshi, V., TA09 I Glazebrook, K.D., TD06 Down, Douglas G., TC04, Glynn, P.W., MB03, TC03, WA08, WA08, WA08 Iftikhar, M., WA05 TD03, WA03 Dube, Parijat, MB07 Ignatiouk-Robert, Irina, WC04 Gnedin, A.V., TD07 Dupuis, Paul, MA03, MA03, Ishizaki, Fumio, TD05 Goel, Ashish , TA09 TB07 Gómez-Corral, A., TC08 Durvy, Mathilde, WC02 Graham, C., TC04 J Grassmann, W.K., WA03 E Gromoll, H.C., MB04, TC04 Jacinto, Gonçalo, TD05 Guillemin, F., TB04, TD03, Jagabathula, S., TA09 Economou, A., MB07, MB07, WA05 Jagers, P., WB09 WC08 Gullu, R., WA09 Jain, Rahul, MB07 Egorova, R., WC05 Günalay, Y., TD09 Janssen, A.J.E.M., TB06, TB06 Eliazar, Iddo, MB09, WA01 Gunawan, Indra, MC03 Jaskiewicz,´ A., MC06 El-Shehawy, S.A., MC08 Gupta, D., TD09 Jelenkovic,´ P.R., TB04 El-Sherbiny, A.A., MC08 Gupta, Varun, MA05 Jennings, O., MB06, TA04, Enders, P., WB07 Gurler, U., MA09 WC09 Ephremides, Anthony, WC02 Gurvich, I., MA04, MB06 Jiang, X., WA04 Es-Saghouani, A., TC08 Jonckheere, Matthieu, WC05 Estrade, Anne, MA02 Jongbloed, G., MA06 Ettl, M., TB09, TB09 H Jouinni, Oualid, WB06 Eveleens, J.L., MA06 Juneja, Sandeep, MA03, TA03, Haas, P.J., MB03 TA03 Hagwood, Charles, TD06 Jung, K., TA04 F Haijema, R., MA07 Hajizadeh, E., WC06 Fan, J., WC07 Hampshire, Robert C., WB07 K Farahani, R.Z., TD09 Hamza, Kais, WB09 Feinberg, E.A., MA07 Hansen, J., MB04 Kaandorp, G.C., WC09 Fernandez, Emmanuel, TC07 Harchol-Balter, Mor, MA05 Kager, W., WA02 Ferreira, F., MC06 Hasenbein, John J., TA04 Kaishev, Vladimir, TB01 Firoozshahian, Amin, TA09 Hautphenne, S., TC06 Kaj, Ingemar, MA02 Fontes, L.R.G., MB02, MB02 He, Q.-M., TC06 Kalra, Himanshu, TA03 Fort, G., WA03 Heidergott, B., TB03, TB03, Kang, Weining, MC04 Foss, S., WB05, WC02 TB03, WC03 Kanta, S., MB07, MB07

164 Kapodistria, S., WC08 Lee, Ju Yong, WB08 Miretskiy, D.I., TB08 Kapuscinski, R., TB09 Lee, N.H., MB04 Mirjafari, K., WC06 Karaesmen, F., TC07, WB06 Lee, Seon Mi, TC05 Miyazawa, M., WA06 Katz, D., TA04 Lee, Yong Hyun, TC05 Mo, S.F., WC01 Kazemnejad, A., WC06 Lehoczky, J., TC04 Moallemi, Ciamac C., TA09 Kekre, Sunder, WA09 Leith, D., WB05 Mohamed, H., MC07 Kella, Offer, WA01 Leizarowitz, Arie, TC07 Möhle, Martin, TB02 Keppo, J., MA01 Lelarge, Marc, MC07, TB04, Momcilovic, P., MB04 Kerner, Yoav, MC08 TD02 Moore, A., WC06 Key, P.B., TA05 Leskelä, L., MC06 Moreno, P., WC08 Khanchi, Aziz, TD04 Lewis, Mark E., WA08 Moriarty, J., MB08 Kim, Bara, WB04 Li, H., TC06, WA06 Mörters, P., WB02 Kim, Ki Baek, WB05 Liese, M., MA01 Morton, David P., TA04 Kim, Kyung Jae, MB05, MB05, Lieshout, P., WC05 Motyer, Allan, WA06 MB05, TC05, TC05 Limic, V., TB02 Moulines, E., WA03 Kim, Sujin, MB03 Lin, Wuqin, MA04 Moyal, P., WA04 Kim, Sunggon, WB08 Lindskog, F., TD01 Mühlethaler, P., TA02 Kim, Tae Ok, MB05, MB05 Litvak, Nelly, TC09, TC09 Müller, A., TD01 King, C., WB05 Liu, J., WC05 Müller, T., WA02 Kirkbride, C., TD06 Liu, Zhen, TB04 Klafter, Joseph, TC02 Löpker, Andreas H., WA05 Koch, I., MB01 Lu, Y., MC05, TB09 N Koijen, Ralph S.J., MB01 Lu, Z.Q. John, TD06 Koksalan, M., WA09 Luczak, M.J., WA02 Najafi, M., TD09 Kolbe, Andreas, WC01 Nakagawa, K., WC04 König, W., WB02 Narayanan, Viswanath C., WC08 Koole, G., MA06, WA08, WC09 M Nazarathy, Y., MB08 Kou, S.G., MA01, TA01 Neininger, R., WB09 Kovaleva, Agnessa, WC04 Machihara, F., MA08 Newman, C.M., MB02, MB02 Kozma, G., WB02 MacPhee, I., TA08 Nijman, Theo E., MB01 Krampe, A., TC01 Magdziarz, M., TC02 Nishimura, S., TD08 Kranenburg, A.A., WA09 Maglaras, Constantinos, MB06 Nobel, R., MA08 Krishnamoorthy, A., WC08 Mahabhashyam, S.R., TB08 Norros, I., WA02 Kroese, D.P., TC03, TC03, TC03 Mahmoodi, S., MA02, TB07 Novikov, A., TA07 Kruk, Lukasz, TC04 Majewski, Kurt, TD04 Nyrhinen, H., TB01 Kuhnt, S., TC01 Malhotra, R., TB08 Kulkarni, Vidyadhar, MB08, Mandjes, M.R.H., TA05, TA05, O TB01 TB08, TB08, TC08, WC05 Kumar, Sunil, MA04, TA09 Margolius, B.H., MB08 O’Neill, P.D., WC06 Kurchan, J., MC02 Massey, Bill, WB07 O’Reilly, Małgorzata, TC06 Kurtz, Thomas G., MB02 Massoulié, Laurent, MC05, Orfão, N.M., WA04, WC03 Kurushima, A., TD07 TB05 Ormeci, E.L., TC07, WB06 Kyprianou, A., MC01 Mazumdar, R.R., TB04, TD03 Oshanin, Gleb, MB09 McCarthy, M.A., WC06 Ozmutlu, Huseyin C., TD09 McDonald, D.R., TB05, TD04, Ozmutlu, Seda, TD09 L TD04, WB05 Meerschaert, M.M., MA02 Lamond, B.F., WB03 Meester, R.J., MB02 P Landfeldt, B., WA05 Mehrotra, Vijay, MA06 Latouche, Guy, TC06, TC06 Meiner, Silke, TD04 Pacheco, A., MC06, TD05, TD08 Leahu, Haralambie, TB03 Merlet, G., WC03 Palmowski, Z., MC01, MC01 Lebedev, D., TC09 Meschenmoser, D., TA02 Papanicolaou, V., WA04 Lee, Chihoon, MC04 Metzler, Ralf, MB09 Park, Jin Soo, MB05, MB05 Lee, J., TD04 Michna, Z., TB01 Park, Wonyoung, WB08

165 12 – Author Index

Pawlak, M., TC01 Roubos, D., MB05 Stolyar, A.L., TA04, WC02 Peköz, Erol A., MC08 Roy, Rahul, TD02 Sudbury, A., WB09 Peng, X.H., MA01 Roychowdhury, Soma, TC01 Sukparungsee, S., TA07 Pérez, J.F., TA07 Rubinstein, R.Y., MC03, MC03, Swaminathan, Jayashankar M., Perry, David, MC09, WA01 TC03, WA03 TB01 Peters, Jan, TB03 Ruiz-Medina, M.D., MA02 Szajowski, K., WC07 Petrou, Evangelia, TD01 Rybko, A.N., WB04 Szekli, R., WA03 Pham, Kim Son, TC09 Piera, F., TD03 Pistorius, Martijn, MC01, MC01 S T Poor, H.V., WC05 Taimre, T., TC03, TC03 Pot, S.A., TA06 Salah Aguir, Mohamed, WB06 Tan, J., TB04 Prabhakar, B., MC05, TA09 Samorodnitsky, G., TD03 Tang, A., MC05 Presman, E.L., TD07 Sanghavi, S., WB09 Taylor, J.W., MA06 Priouret, P., WA03 Scandolo, G., MA01 Taylor, Peter, TC06, WA06, Proutière, A., TB05, WC05 Schaal, Stefan, TB03 WC06 Puchała, Z., MC01 Scheffler, H.P., MA02 Tekin, E., WC09 Puha, A.L., TC04 Scheinhardt, W.R.W., TA03, Tezcan, Tolga, MA04, MB06 TB08, TB08, TB08, TC09 Theodorou, Evangelos, TB03 Scheller-Wolf, A., TB06, WA09, Thiran, Patrick, WC02 Q WB08 Thönnes, Elke, TA02 Scherer, Matthias, MB01 Qiu, Minmin, MA09, MC03 Tibi, D., TB05 Schertzer, E., MB02, MB02 Quinn, J., WA07 Tijms, H.C., TD08 Schmidt, V., TA02 Timmer, J., MA09 Schweinsberg, Jason, TB02 Tokarev, D., WB09 Secomandi, Nicola, WA09 R Torrisi, G.L., TD02 Segers, Johan, TA07 Touati, Corinne, MB07 Radunovic,´ B., TD02 Serfozo, R.F., MC08 Towsley, Don, TB04 Ramanan, K., MB04, MC04, Sethi, S., WA03 Toyoizumi, Hiroshi, TC08 MC07, TC04, TC04 Sethi, S.P., MA01 Travani, Foad, WB08 Ramírez-Hernández, José A., Shah, D., MC05, TA09, WB09 Twigg, Andy, TB05 TC07 Sheu, S.J., WC04 Randhawa, R.S., MA04 Shin, Yang Woo, MA08 Rataj, J., TA02 Shlosman, S.B., WB04 V Ravishankar, K., MB02, MB02 Shneer, V., WC02 Redig, F., MC02 Shorten, R., WB05 Valdivieso, Luis, MB01 Reed, J.E., TA04, TB06 Shreve, S., MC07, TC04 Van den Berg, E., TD09 Reiman, Martin I., MB04 Shwartz, Adam, TC07 Van den Berg, J., MC02 Reittu, H., WA02 Sidorova, N., WB02 Van den Berg, J.L., TA05, TB08 Remiche, M.-A., TC06 Simatos, F., TD05 Van den Schrieck, Ren, Changrui, MA09, MC03 Singh, T., WA05 Jean-Christophe, TA06, WB06 Resing, J.A.C., TA08, TC05 Siris, Vasilios A., TA05 Van der Hofstad, R., MC02, Reynolds, C., MB04 Snyder, R.D., MA06 TC09, WA02, WA02 Riaño, G., MC08, TA07 Sokolov, Igor M., TC02 Van der Laan, D.A., MA07 Ribeiro, H., TD08 Soltani, A.R., MA02, TB07 Van der Mei, R.D., MB05, TA08 Ridder, A., TC03 Son, Jung Je, TC05, TC05, TC05 Van der Wal, Jan, MA07, TC07 Righter, R., WA08 Sonin, Isaac, TD07, WA07 Van Doorn, E.A., MC06 Robert, Ph., TB05, TD05, WA05 Sourirajan, K., TB09 Van Houtum, G.J., MA09, Roijers, Frank, TA05 Spencer, S.E.F., WC06 WA09, WA09, WB07 Rolski, T., MC01 Spodarev, E., TA02 Van Leeuwaarden, Johan S.H., Rosenberg, C.P., TB04 Squillante, M.S., TB09, WA06, TB06, TB06, WA05, WA06, Ross, Kevin J., TB07 WA06 WB02 Ross, Sheldon, MC08 Stadje, Wolfgang, WA01 Van Lierde, Sarah, TC06 Rothblum, U.G., MA07 Steckley, Samuel G., MA06 Van Lieshout, M.N.M., TA02

166 Van Roy, Benjamin, TA09 Wierman, Adam, MA05, MA05 Yechiali, Uri, TA08 Vázquez-Abad, F., WC03 Williams, R.J., MB04, TB07 Yi, Y., WC05 Veatch, Michael, WB03 Willsky, A., WB09 Yin, Wenjun, MC09, TC01, Vesilo, R.A., TB06, WB08 Winands, E.M.M., MA05, TA08, WC03 Vladimirov, A.A., WB04 WA06 Ying, Yu, TB04 Vlasiou, Maria, TA08, WC08 Wirth, F., WB05 Yuan, Taoying, TB03 Vliegen, I.M.H., WA09 Wischik, D., WB05 Yudaeva, N.V., MC07 Vojnovic, Milan, TB05 Volk-Makarewicz, W., WC03 Volkovich, Y., TC09 X Z Xia, Cathy H., TB04 Zachary, S., MB04 Xia, Li, WC03 Zagst, Rudi, WC01 W Xiao, Yongbo, MC09 Zazanis, Michael, WA04 Xie, Ming, MC09 Wang, Qinhua, MC03 Zeevi, Assaf, MA04, TA03, Xiu, Cathy, WB07 Wang, Wei, MA09, MC03 TA06 Xu, H., TB09 Wang, Xiaofeng (Mulan), WA09 Zhan, Li, TA03 Ward, A.R., MB06 Zhang, Bin, MC09 Weber, R.R., WC07 Y Zhao, Y.Q., TC06, WA06 Weber, S., MA01 Zheng, S., TB09 Weerasinghe, A., MC04 Yannacopoulos, A.N., WA04 Zitkovic,´ Gordan, WC01 Weiss, G., MB08 Yao, D.D., MC09, TB09 Ziya, S., WC09 Werker, Bas J.M., MB01 Yatomi, N., TD08 Zwart, Bert, MA05, TB06, TB06, Weron, Aleksander, TC02 Yau, K.K.W., WC01 TD03, WC05

167 12 – Author Index

168 Tuesday July 10, 2007 Track 1 Track 2 Track 3 Track 4 Track 5 Track 6 Track 7 Track 8 Track 9 Room CZ 4 CZ 5 CZ 10 CZ 11 CZ 12 CZ 13 CZ 14 CZ 15 CZ 16 TA Financial Stochastic Rare Event Scheduling Ad Hoc Call Centers 2 Statistics 1 Polling Switch Scheduling 8:30am - 10:00am Engineering 3 Geometry Simulation 2 and Stability Networks Models Kou Van Lieshout Juneja Hasenbein Van den Berg Koole Segers Winands Moallemi

Giesecke Spodarev De Boer Jennings Siris Helber Areepong MacPhee Bayati Chen Van Lieshout Bassamboo Stolyar Roijers Pot Sukparungsee Yechiali Prabhakar Cont Thönnes Blanchet Gamarnik Gibbens Van den Schrieck Pérez Resing Shah Cai Blaszczyszyn Juneja Hasenbein Bekker Bassamboo Segers Winands Moallemi TB Finance and Ruin Random Gradient Asymptotic Resource Many-server Control and Fluid Models Supply Chains 10:30am - 12:00pm Partitions Estimation Analysis 2 Control Models 2 PDE Collamore Gnedin Heidergott Xia Borst Van Leeuwaarden Atar Scheinhardt Ettl / De Kok / Zwart Ballotta Berestycki Peters Jelenkovic´ Massoulié Van Leeuwaarden Budhiraja Gautam Squillante Michna Bogachev Leahu Mazumdar Proutière Zwart Dupuis Malhotra Yao Collamore Möhle Yuan Chaintreau Robert Reed Atar Scheinhardt Ettl Kulkarni Schweinsberg Heidergott Xia Vojnovic Scheller-Wolf Soltani Miretskiy De Kok 12:20pm - 1:20pm Tutorial Paul Glasserman 1:30pm - 2:30pm Plenary - Alain-Sol Sznitman (Room: Blauwe Zaal) TC Statistics 2 Anomalous Cross-entropy Measure-val. Communication Matrix-Analytic MDP 2 Queueing World Wide Web 2:45pm - 4:15pm Diffusion Processes Systems 2 Methods Models 2 Kuhnt Metzler Kroese Puha Choi Squillante / Örmeci Gómez- Litvak Taylor Corral Bai Klafter Rubinstein Ramanan Beekhuizen Zhao Al-Ibrahim Blanc Lebedev Bhattacharya Barkai Botev Lehoczky Kim Hautphenne Shwartz Es-Saghouani Van der Hofstad Pawlak Sokolov Taimre Graham Baek O’Reilly Fernandez Toyoizumi Volkovich Kuhnt Weron Ridder Puha Choi Latouche Örmeci Gómez-Corral Litvak TD Lévy Processes Spatial Asymptotics Large Computer- Applied Optimal Queueing in Engineering 4:45pm - 6:15pm and Finance Models 1 Deviations 1 comm. Probability Stopping 1 Telecom Rieder / Bäuerle Foss / Blanchet / Majewski Hwang Hagwood Gnedin Tijms Günalay Stolyar Mikosch Petrou Torrisi Blanchet Lee Simatos Coakley Kurushima Nishimura Van den Berg Lindskog Blaszczyszyn Hult Khanchi Jacinto Lu Helmes Garcia Farahani Bäuerle Roy Zwart Meiner Hwang Hagwood Gnedin Ribeiro Ozmutlu Lelarge Mazumdar Majewski Ding Presman Tijms Günalay 6:15pm - 7:00pm APS Business Meeting (Location: CZ 4) 6:30pm - 8:00pm Social Event (Location: outside the Auditorium) Wednesday July 11, 2007 Track 1 Track 2 Track 3 Track 4 Track 5 Track 6 Track 7 Track 8 Track 9 Room CZ 4 CZ 5 CZ 10 CZ 11 CZ 12 CZ 13 CZ 14 CZ 15 CZ 16 WA Queues and Random Graphs Simulation 2 Networks Internet Infinite Multidim. Optimal Flexible Inventory 3 8:30am - 10:00am Lévy Processes Markov Chains Stopping 2 Servers Kella Norros Grassmann Zazanis Fricker Squillante / Sonin Down Van Houtum Taylor Perry Hooghiemstra Awad Moyal Löpker Miyazawa Belenky Ayhan Vliegen Eliazar Müller Fort Zazanis Caglar˜ Gamarnik Quinn Lewis Gullu Stadje Luczak Grassmann Orfão Fricker Taylor Sonin Righter Scheller-Wolf Kella Norros Szekli Jiang Squillante Down Van Houtum WB Last Minute Self-interacting Stoch. Prog. Fluid Limits and TCP Call Centers 3 Control for Queueing Combinatorial 10:30am - 12:00pm Talks Stoch. Processes & Opt. 2 Mean Fields Service Models 3 Structures Boxma Van der Hofstad Veatch D’Auria Baccelli Koole Hampshire Brandt Gamarnik / Ramanan Den Iseger Holmes Burnetas D’Auria Wirth Chevalier Massey Vesilo Neininger Sidorova Hodge Vladimirov Wischik Jouini Enders Kim Baryshnikov Kozma Lamond Kim Carofiglio Örmeci Xiu Balcıoglu˜ Hamza Van Leeuwaarden Veatch McDonald Bhulai Hampshire Brandt Shah 12:20pm - 1:20pm Tutorial Maury Bramson WC Risk and Spatial Models 2 Perturb. & Large Bandwidth Biology Games 2 Queueing Health Care 1:30pm - 3:00pm Insurance Disc. Events Deviations 2 Sharing Models 4 Kolbe Foss / Stolyar Heidergott Ignatiouk-Robert Borst Spencer Weber Vlasiou Ziya

Biffis Thiran Merlet Chiang Egorova Kazemnejad Fan Kapodistria Kaandorp Mo Stolyar Orfão Kovaleva Jonckheere Moore Szajowski Moreno Jennings Zitkovic´ Shneer Xia Ignatiouk-Robert Lieshout Spencer Weber Krishnamoorthy Tekin Kolbe Ephremides Heidergott Nakagawa Liu Vlasiou Ziya 3:15pm - 4:15pm Plenary - Frank Kelly (Room: Blauwe Zaal)