An Application of Matrix Analytic Methods to Queueing Models with Polling
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Fluid M/M/1 Catastrophic Queue in a Random Environment
RAIRO-Oper. Res. 55 (2021) S2677{S2690 RAIRO Operations Research https://doi.org/10.1051/ro/2020100 www.rairo-ro.org FLUID M=M=1 CATASTROPHIC QUEUE IN A RANDOM ENVIRONMENT Sherif I. Ammar1;2;∗ Abstract. Our main objective in this study is to investigate the stationary behavior of a fluid catas- trophic queue of M=M=1 in a random multi-phase environment. Occasionally, a queueing system expe- riences a catastrophic failure causing a loss of all current jobs. The system then goes into a process of repair. As soon as the system is repaired, it moves with probability qi ≥ 0 to phase i. In this study, the distribution of the buffer content is determined using the probability generating function. In addition, some numerical results are provided to illustrate the effect of various parameters on the distribution of the buffer content. Mathematics Subject Classification. 90B22, 60K25, 68M20. Received January 12, 2020. Accepted September 8, 2020. 1. Introduction In recent years, studies on queueing systems in a random environment have become extremely important owing to their widespread application in telecommunication systems, advanced computer networks, and manufacturing systems. In addition, studies on fluid queueing systems are regarded as an important class of queueing theory; the interpretation of the behavior of such systems helps us understand and improve the behavior of many applications in our daily life. A fluid queue is an input-output system where a continuous fluid enters and leaves a storage device called a buffer; the system is governed by an external stochastic environment at randomly varying rates. These models have been well established as a valuable mathematical modeling method and have long been used to estimate the performance of certain systems as telecommunication systems, transportation systems, computer networks, and production and inventory systems. -
The MVA Priority Approximation
The MVA Priority Approximation RAYMOND M. BRYANT and ANTHONY E. KRZESINSKI IBM Thomas J. Watson Research Center and M. SEETHA LAKSHMI and K. MANI CHANDY University of Texas at Austin A Mean Value Analysis (MVA) approximation is presented for computing the average performance measures of closed-, open-, and mixed-type multiclass queuing networks containing Preemptive Resume (PR) and nonpreemptive Head-Of-Line (HOL) priority service centers. The approximation has essentially the same storage and computational requirements as MVA, thus allowing computa- tionally efficient solutions of large priority queuing networks. The accuracy of the MVA approxima- tion is systematically investigated and presented. It is shown that the approximation can compute the average performance measures of priority networks to within an accuracy of 5 percent for a large range of network parameter values. Accuracy of the method is shown to be superior to that of Sevcik's shadow approximation. Categories and Subject Descriptors: D.4.4 [Operating Systems]: Communications Management-- network communication; D.4.8 [Operating Systems]: Performance--modeling and prediction; queuing theory General Terms: Performance, Theory Additional Key Words and Phrases: Approximate solutions, error analysis, mean value analysis, multiclass queuing networks, priority queuing networks, product form solutions 1. INTRODUCTION Multiclass queuing networks with product-form solutions [3] are widely used to model the performance of computer systems and computer communication net- works [11]. The effective application of these models is largely due to the efficient computational methods [5, 9, 13, 18, 21] that have been developed for the solution of product-form queuing networks. However, many interesting and significant system characteristics cannot be modeled by product-form networks. -
Official Journal of the Bernoulli Society for Mathematical Statistics And
Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty One Number Four November 2015 ISSN: 1350-7265 CONTENTS Papers BEN ALAYA, M., HAJJI, K. and KEBAIER, A. 1947 Importance sampling and statistical Romberg method REBELLES, G. 1984 Pointwise adaptive estimation of a multivariate density under independence hypothesis OGIHARA, T. 2024 Local asymptotic mixed normality property for nonsynchronously observed diffusion processes ADAMCZAK, R. and BEDNORZ, W. 2073 Some remarks on MCMC estimation of spectra of integral operators AKASHI, F., LIU, Y. and TANIGUCHI, M. 2093 An empirical likelihood approach for symmetric α-stable processes VOS, P. and WU, Q. 2120 Maximum likelihood estimators uniformly minimize distribution variance among distribution unbiased estimators in exponential families BOURGUIN, S. 2139 Poisson convergence on the free Poisson algebra FANG, X. and RÖLLIN, A. 2157 Rates of convergence for multivariate normal approximation with applications to dense graphs and doubly indexed permutation statistics CHONG, C. and KLÜPPELBERG, C. 2190 Integrability conditions for space–time stochastic integrals: Theory and applications CHEN, F. and NKURUNZIZA, S. 2217 Optimal method in multiple regression with structural changes GIACOMIN, G. and MERLE, M. 2242 Weak noise and non-hyperbolic unstable fixed points: Sharp estimates on transit and exit times BULL, A.D. 2289 Adaptive-treed bandits (continued) The papers published in Bernoulli are indexed or abstracted in Current Index to Statistics, Mathematical Reviews, Statistical Theory and Method Abstracts-Zentralblatt (STMA-Z), and Zentralblatt für Mathematik (also avalaible on the MATH via STN database and Compact MATH CD-ROM). A list of forthcoming papers can be found online at http://www. -
Queueing Theory and Process Flow Performance
QUEUEING THEORY AND PROCESS FLOW PERFORMANCE Chang-Sun Chin1 ABSTRACT Queuing delay occurs when a number of entities arrive for services at a work station where a server(s) has limited capacity so that the entities must wait until the server becomes available. We see this phenomenon in the physical production environment as well as in the office environment (e.g., document processing). The obvious solution may be to increase the number of servers to increase capacity of the work station, but other options can attain the same level of performance improvement. The study selects two different projects, investigates their submittal review/approval process and uses queuing theory to determine the major causes of long lead times. Queuing theory provides good categorical indices—variation factor, utilization factor and process time factor—for diagnosing the degree of performance degradation from queuing. By measuring the magnitude of these factors and adjusting their levels using various strategies, we can improve system performance. The study also explains what makes the submittal process of two projects perform differently and suggests options for improving performance in the context of queuing theory. KEY WORDS Process time, queueing theory, submittal, variation, utilization INTRODUCTION Part of becoming lean is eliminating all waste (or muda in Japanese). Waste is “any activity which consumes resources but creates no value” (Womack and Jones 2003). Waiting, one of seven wastes defined by Ohno of Toyota (Ohno 1988), can be seen from two different views: “work waiting” or “worker waiting.” “Work waiting” occurs when servers (people or equipment) at work stations are not available when entities (jobs, materials, etc) arrive at the work stations, that is, when the servers are busy and entities wait in queue. -
Convoluted Fractional Poisson Process
ALEA, Lat. Am. J. Probab. Math. Stat. 18, 1241–1265 (2021) DOI: 10.30757/ALEA.v18-46 Convoluted Fractional Poisson Process Kuldeep Kumar Kataria and Mostafizar Khandakar Department of Mathematics Indian Institute of Technology Bhilai Raipur-492015, India. E-mail address: [email protected], [email protected] Abstract. In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). It has integer valued jumps of size j ≥ 1. The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. A special case of the CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a Lévy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property. 1. Introduction The Poisson process is a renewal process with exponentially distributed waiting times. This Lévy process is often used to model the counting phenomenon. Empirically, it is observed that the process with heavy-tailed distributed waiting times offers a better model than the ones with light-tailed distributed waiting times. For this purpose several fractional generalizations of the homogeneous Poisson process are introduced and studied by researchers in the past two decades. -
Text Mining and Ruin Theory: a Case Study of Research on Risk Models with Dependence ∗
Text mining and ruin theory: A case study of research on risk models with dependence ∗ Authors: Renata G. Alcoforado { ISEG & CEMAPRE, Universidade de Lisboa; Department of Accounting and Actuarial Sciences, Universidade Federal dePernambuco, Portugal & Brazil ([email protected]) Alfredo D. Eg´ıdio dos Reis { ISEG & CEMAPRE, Universidade de Lisboa Portugal ([email protected]) Abstract: • This paper aims to analyze unstructured data using a text mining approach. The work was motivated in order to organize and structure research in Risk Theory. In our study, the subject to be analyzed is composed by 27 published papers of the risk and ruin theory topic, area of actuarial science. They were coded into 32 categories. For the purpose, all data was analyzed and figures were produced using the software NVivo 11 plus. Software NVivo is a specialized tool in analyzing unstructured data, although it is commonly used just for qualitative research. We used it for Quali-Quant analysis. Key-Words: • Big data; Unstructured data; Text mining; Risk theory; Ruin theory; Dependence modeling. AMS Subject Classification: • 49A05, 78B26. ∗The opinions expressed in this text are those of the authors and do not necessarily reflect the views of any organization. 2 Renata G. Alcoforado and Alfredo D. Eg´ıdiodos Reis Using the \revstat.sty" Package 3 1. INTRODUCTION As widely known, Big Data is an area of great development in statistics. We can define Big Data as \a phenomenon defined by the rapid acceleration in the expanding volume of high velocity, complex, and diverse types of data. Big Data is often defined along three dimensions { volume, velocity, and variety" (TechAmerica Foundation's Federal Big Data Commission, 2012). -
M/M/C Queue with Two Priority Classes
Submitted to Operations Research manuscript (Please, provide the mansucript number!) M=M=c Queue with Two Priority Classes Jianfu Wang Nanyang Business School, Nanyang Technological University, [email protected] Opher Baron Rotman School of Management, University of Toronto, [email protected] Alan Scheller-Wolf Tepper School of Business, Carnegie Mellon University, [email protected] This paper provides the first exact analysis of a preemptive M=M=c queue with two priority classes having different service rates. To perform our analysis, we introduce a new technique to reduce the 2-dimensionally (2D) infinite Markov Chain (MC), representing the two class state space, into a 1-dimensionally (1D) infinite MC, from which the Generating Function (GF) of the number of low-priority jobs can be derived in closed form. (The high-priority jobs form a simple M=M=c system, and are thus easy to solve.) We demonstrate our methodology for the c = 1; 2 cases; when c > 2, the closed-form expression of the GF becomes cumbersome. We thus develop an exact algorithm to calculate the moments of the number of low-priority jobs for any c ≥ 2. Numerical examples demonstrate the accuracy of our algorithm, and generate insights on: (i) the relative effect of improving the service rate of either priority class on the mean sojourn time of low-priority jobs; (ii) the performance of a system having many slow servers compared with one having fewer fast servers; and (iii) the validity of the square root staffing rule in maintaining a fixed service level for the low priority class. -
Product-Form in Queueing Networks
Product-form in queueing networks VRIJE UNIVERSITEIT Product-form in queueing networks ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Vrije Universiteit te Amsterdam, op gezag van de rector magnificus dr. C. Datema, hoogleraar aan de faculteit der letteren, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der economische wetenschappen en econometrie op donderdag 21 mei 1992 te 15.30 uur in het hoofdgebouw van de universiteit, De Boelelaan 1105 door Richardus Johannes Boucherie geboren te Oost- en West-Souburg Thesis Publishers Amsterdam 1992 Promotoren: prof.dr. N.M. van Dijk prof.dr. H.C. Tijms Referenten: prof.dr. A. Hordijk prof.dr. P. Whittle Preface This monograph studies product-form distributions for queueing networks. The celebrated product-form distribution is a closed-form expression, that is an analytical formula, for the queue-length distribution at the stations of a queueing network. Based on this product-form distribution various so- lution techniques for queueing networks can be derived. For example, ag- gregation and decomposition results for product-form queueing networks yield Norton's theorem for queueing networks, and the arrival theorem implies the validity of mean value analysis for product-form queueing net- works. This monograph aims to characterize the class of queueing net- works that possess a product-form queue-length distribution. To this end, the transient behaviour of the queue-length distribution is discussed in Chapters 3 and 4, then in Chapters 5, 6 and 7 the equilibrium behaviour of the queue-length distribution is studied under the assumption that in each transition a single customer is allowed to route among the stations only, and finally, in Chapters 8, 9 and 10 the assumption that a single cus- tomer is allowed to route in a transition only is relaxed to allow customers to route in batches. -
Italic Entries Indicatefigures. 319-320,320
Index A Analytic hierarchy process, 16-24, role of ORIMS, 29-30 17-18 Availability,31 A * algorithm, 1 absolute, relative measurement Averch-lohnson hypothesis, 31 Acceptance sampling, 1 structural information, 17 Accounting prices. 1 absolute measurement, 23t, 23-24, Accreditation, 1 24 B Active constraint, 1 applications in industry, Active set methods, 1 government, 24 Backward chaining, 33 Activity, 1 comments on costlbenefit analysis, Backward Kolmogorov equations, 33 Activity-analysis problem, 1 17-19 Backward-recurrence time, 33 Activity level, 1 decomposition of problem into Balance equations, 33 Acyclic network, 1 hierarchy, 21 Balking, 33 Adjacency requirements formulation, eigenvector solution for weights, Banking, 33-36 facilities layout, 210 consistency, 19-20, 20t banking, 33-36 Adjacent, 1 employee evaluation hierarchy, 24 portfolio diversification, 35-36 Adjacent extreme points, 1 examples, 21-24 portfolio immunization, 34-35 Advertising, 1-2, 1-3 fundamental scale, 17, 18, 18t pricing contingent cash flows, competition, 3 hierarchic synthesis, rank, 20-21 33-34 optimal advertising policy, 2-3 pairwise comparison matrix for BarChart, 37 sales-advertising relationship, 2 levell, 21t, 22 Barrier, distance functions, 37-39 Affiliated values bidding model, 4 random consistency index, 20t modified barrier functions, 37-38 Affine-scaling algorithm, 4 ranking alternatives, 23t modified interior distance Affine transformation, 4 ranking intensities, 23t functions, 38-39 Agency theory, 4 relative measurement, 21, 21-22t, Basic feasible solution, 40 Agriculture 21-24 Basic solution, 40 crop production problems at farm structural difference between Basic vector, 40, 41 level,429 linear, nonlinear network, 18 Basis,40 food industry and, 4-6 structuring hierarchy, 20 Basis inverse, 40 natural resources, 428-429 synthesis, 23t Batch shops, 41 regional planning problems, 429 three level hierarchy, 17 Battle modeling, 41-44 AHP, 7 Animation, 24 attrition laws, 42-43 AI, 7. -
Lecture 5/6 Queueing
6.263/16.37: Lectures 5 & 6 Introduction to Queueing Theory Eytan Modiano Massachusetts Institute of Technology Eytan Modiano Slide 1 Packet Switched Networks Messages broken into Packets that are routed To their destination PS PS PS Packet Network PS PS PS Buffer Packet PS Switch Eytan Modiano Slide 2 Queueing Systems • Used for analyzing network performance • In packet networks, events are random – Random packet arrivals – Random packet lengths • While at the physical layer we were concerned with bit-error-rate, at the network layer we care about delays – How long does a packet spend waiting in buffers ? – How large are the buffers ? • In circuit switched networks want to know call blocking probability – How many circuits do we need to limit the blocking probability? Eytan Modiano Slide 3 Random events • Arrival process – Packets arrive according to a random process – Typically the arrival process is modeled as Poisson • The Poisson process – Arrival rate of λ packets per second – Over a small interval δ, P(exactly one arrival) = λδ + ο(δ) P(0 arrivals) = 1 - λδ + ο(δ) P(more than one arrival) = 0(δ) Where 0(δ)/ δ −> 0 as δ −> 0. – It can be shown that: (!T)n e"!T P(n arrivals in interval T)= n! Eytan Modiano Slide 4 The Poisson Process (!T)n e"!T P(n arrivals in interval T)= n! n = number of arrivals in T It can be shown that, E[n] = !T E[n2 ] = !T + (!T)2 " 2 = E[(n-E[n])2 ] = E[n2 ] - E[n]2 = !T Eytan Modiano Slide 5 Inter-arrival times • Time that elapses between arrivals (IA) P(IA <= t) = 1 - P(IA > t) = 1 - P(0 arrivals in time -
(Thesis Proposal) Takayuki Osogami
Resource Allocation Solutions for Reducing Delay in Distributed Computing Systems (Thesis Proposal) Takayuki Osogami Department of Computer Science Carnegie Mellon University [email protected] April, 2004 Committee members: Mor Harchol-Balter (Chair) Hui Zhang Bruce Maggs Alan Scheller-Wolf (Tepper School of Business) Mark Squillante (IBM Research) 1 Introduction Waiting time (delay) is a source of frustration for users who receive service via computer or communication systems. This frustration can result in lost revenue, e.g., when a customer leaves a commercial web site to shop at a competitor's site. One obvious way to decrease delay is simply to buy (more expensive) faster machines. However, we can also decrease delay for free with given resources by making more efficient use of resources and by better scheduling jobs (i.e., by changing the order of jobs to be processed). For single server systems, it is well understood how to minimize mean delay, namely by the shortest remaining processing time first (SRPT) scheduling policy. SRPT can provide mean delay an order of magnitude smaller than a naive first come first serve (FCFS) scheduling policy. Also, the mean delay under various scheduling policies, including SRPT and FCFS, can be easily analyzed for a relatively broad class of single server systems (M/GI/1 queues). However, utilizing the full potential computing power of multiserver systems and analyzing their performance are much harder problems than for the case of a single server system. Despite the ubiquity of multiserver systems, it is not known how we should assign jobs to servers and how we should schedule jobs within each server to minimize the mean delay in multiserver systems. -
Steady-State Analysis of Reflected Brownian Motions: Characterization, Numerical Methods and Queueing Applications
STEADY-STATE ANALYSIS OF REFLECTED BROWNIAN MOTIONS: CHARACTERIZATION, NUMERICAL METHODS AND QUEUEING APPLICATIONS a dissertation submitted to the department of mathematics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy By Jiangang Dai July 1990 c Copyright 2000 by Jiangang Dai All Rights Reserved ii I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. J. Michael Harrison (Graduate School of Business) (Principal Adviser) I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. Joseph Keller I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. David Siegmund (Statistics) Approved for the University Committee on Graduate Studies: Dean of Graduate Studies iii Abstract This dissertation is concerned with multidimensional diffusion processes that arise as ap- proximate models of queueing networks. To be specific, we consider two classes of semi- martingale reflected Brownian motions (SRBM's), each with polyhedral state space. For one class the state space is a two-dimensional rectangle, and for the other class it is the d general d-dimensional non-negative orthant R+. SRBM in a rectangle has been identified as an approximate model of a two-station queue- ing network with finite storage space at each station.