Fluid Limits for Multiclass Many Server Queues with General Reneging Distributions and Head-Of-Line Scheduling
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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. -
Extremes and Fluid Queues
UvA-DARE (Digital Academic Repository) Extremes and fluid queues Dieker, A.B. Publication date 2006 Document Version Final published version Link to publication Citation for published version (APA): Dieker, A. B. (2006). Extremes and fluid queues. Ponsen & Looijen. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:30 Sep 2021 Extremes and uid queues Extremes and uid queues / Antonius Bernardus Dieker, 2006 Proefschrift Universiteit van Amsterdam Met lit. opg. { Met samenvatting in het Nederlands Omslagontwerp door Tobias Baanders Gedrukt door Ponsen & Looijen BV ISBN 90-5776-151-3 Dit onderzoek kwam tot stand met steun van NWO Extremes and uid queues ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magni¯cus prof. -
Heterogeneous Arrival Fluid Model Driven by an M/M/1 Queue with Exponential Vacation Subject to Catastrophe M
The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 Heterogeneous Arrival Fluid Model Driven by an M/M/1 Queue with Exponential Vacation Subject to Catastrophe 1 2 M. Deepa , Dr. K. Julia Rose Mary 1Research Scholar, Nirmala College For Women 2Asso.Prof.of Mathematics, Nirmala College For Women [email protected] 2 [email protected] Abstract- Fluid model is suitable for modeling traffic network where individual arrival has less impact on the performance of the network. For evaluating the performance measure, it is important to obtain information about the buffer content. This paper studies a fluid model driven by an M/M/1 queue with heterogeneous arrival and exponential vacation subject to catastrophe under steady state conditions. Based on that explicit expression for the Laplace transform of the stationary buffer content distribution with the minimal positive solution to a crucial quadratic equation is obtained .Then the performance measure – mean of the buffer content which is dependent on vacation parameter is obtained .Finally, the parameters effect on the mean buffer content is also illustrated by sensitivity analysis. Keywords- Fluid queue, Heterogeneous arrival, Exponential vacation, Catastrophe and Buffer Content. I. INTRODUCTION A Fluid queue is a mathematical model used to describe fluid level in a reservoir subject to randomly determined periods of filling and emptying the system without interruption called a buffer, according to a randomly varying rate regulated by an external stochastic environment. Such fluid queues are used as a mathematical tool for modeling, for example, to approximate discrete models, model the spread of wildfires in ruin theory and to model high speed data networks, a router, computer networks including call admission control, traffic shaping and modeling of TCP and production and inventory systems. -
Informs 2007 Proceedings
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 -
Markov-Modulated and Feedback Fluid Queues
Markov-modulated and feedback fluid queues Werner Scheinhardt Faculty of Mathematical Sciences University of Twente P.O. Box 217 7500 AE Enschede The Netherlands ISBN 90-3651248-4 MARKOV-MODULATED AND FEEDBACK FLUID QUEUES PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof. dr. F.A. van Vught, volgens besluit van het College voor Promoties in het openbaar te verdedigen op vrijdag 4 december 1998 te 15.00 uur. door Willem Richard Werner Scheinhardt geboren op 24 februari 1969 te Santiago Dit proefschrift is goedgekeurd door de promotor en de assistent promotor, prof. dr. ir. J.H.A. de Smit dr. ir. E.A. van Doorn Voorwoord Aan het eind van dit proefschrift gekomen, rest nog het schrijven van het begin ervan, het voorwoord. Gebruikelijk is om daarin allen te bedanken die op ´e´en of andere wijze aan de totstandkoming van het proefschrift hebben bijgedragen. Graag houd ik deze traditie in ere, en wel omdat de volgende personen dit ten volle verdienen. Uiteraard wil ik beginnen met mijn dagelijks begeleider Erik van Doorn hartelijk te bedanken voor zijn inzet en enthousiasme. Van v´o´or het eerste sollicitatiegesprek tot na het laatste 2-teken was hij intensief betrokken bij mijn doen en laten. Een belangrijk deel van het onderzoek in dit proefschrift is in samenwerking met hem tot stand gekomen. Ook de samenwerking met Dick Kroese was aangenaam en productief. Hij wist vaak antwoorden die ik zelf niet zou hebben gevonden, en leerde mij omgaan met het symbolisch manipulatie-pakket \Mathematica", dat heel wat \tiresome analysis" heeft uitgevoerd. -
EUROPEAN CONFERENCE on QUEUEING THEORY 2016 Urtzi Ayesta, Marko Boon, Balakrishna Prabhu, Rhonda Righter, Maaike Verloop
EUROPEAN CONFERENCE ON QUEUEING THEORY 2016 Urtzi Ayesta, Marko Boon, Balakrishna Prabhu, Rhonda Righter, Maaike Verloop To cite this version: Urtzi Ayesta, Marko Boon, Balakrishna Prabhu, Rhonda Righter, Maaike Verloop. EUROPEAN CONFERENCE ON QUEUEING THEORY 2016. Jul 2016, Toulouse, France. 72p, 2016. hal- 01368218 HAL Id: hal-01368218 https://hal.archives-ouvertes.fr/hal-01368218 Submitted on 19 Sep 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EUROPEAN CONFERENCE ON QUEUEING THEORY 2016 Toulouse July 18 – 20, 2016 Booklet edited by Urtzi Ayesta LAAS-CNRS, France Marko Boon Eindhoven University of Technology, The Netherlands‘ Balakrishna Prabhu LAAS-CNRS, France Rhonda Righter UC Berkeley, USA Maaike Verloop IRIT-CNRS, France 2 Contents 1 Welcome Address 4 2 Organization 5 3 Sponsors 7 4 Program at a Glance 8 5 Plenaries 11 6 Takács Award 13 7 Social Events 15 8 Sessions 16 9 Abstracts 24 10 Author Index 71 3 1 Welcome Address Dear Participant, It is our pleasure to welcome you to the second edition of the European Conference on Queueing Theory (ECQT) to be held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT in Toulouse. -
A Lévy-Driven Stochastic Queueing System with Server Breakdowns and Vacations
mathematics Article A Lévy-Driven Stochastic Queueing System with Server Breakdowns and Vacations Yi Peng 1 and Jinbiao Wu 2,* 1 School of Mathematical Science, Changsha Normal University, Changsha 410100, China; mailto:[email protected] 2 School of Mathematics and Statistics, Central South University, Changsha 410083, China * Correspondence: [email protected] Received: 24 June 2020; Accepted: 28 July 2020; Published: 29 July 2020 Abstract: Motivated by modelling the data transmission in computer communication networks, we study a Lévy-driven stochastic fluid queueing system where the server may subject to breakdowns and repairs. In addition, the server will leave for a vacation each time when the system is empty. We cast the workload process as a Lévy process modified to have random jumps at two classes of stopping times. By using the properties of Lévy processes and Kella–Whitt martingale method, we derive the limiting distribution of the workload process. Moreover, we investigate the busy period and the correlation structure. Finally, we prove that the stochastic decomposition properties also hold for fluid queues with Lévy input. Keywords: fluid queueing systems; Lévy processes; martingales; queues with Lévy input; server breakdowns and vacations; stochastic decomposition 1. Introduction Lévy processes are very important stochastic processes with stationary and independent increments. They are one of the most important families of stochastic processes arising in many areas of probability theory and stochastic operations research. They cover a number of very important and well-studied stochastic processes as special cases, such as Brownian motion, compound Poisson processes, stable processes and subordinators. Lévy processes are widely used as models in the study of stochastic service systems, insurance risks, stochastic inventory systems, reliability engineering, computer communication systems, stochastic finance systems and so on. -
An Analytical Model Driven by Fluid Queue for Battery Life Time of a User Equipment in LTE-A Networks ∗ Anupam Gautam A,S
Physical Communication 30 (2018) 213–219 Contents lists available at ScienceDirect Physical Communication journal homepage: www.elsevier.com/locate/phycom Full length article An analytical model driven by fluid queue for battery life time of a user equipment in LTE-A networks ∗ Anupam Gautam a,S. Dharmaraja b, a Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India b Bharti School of Telecommunication Technology and Management, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India article info a b s t r a c t Article history: During the Discontinuous Reception (DRX) mechanism, a User Equipment (UE) turns off most of its Received 13 October 2017 components for a prolonged period. The battery charge of a UE, which is a continuous quantity, arises Received in revised form 28 May 2018 as a fluid model. In this scenario, we represent the system as the reservoir where battery charge gets Accepted 13 August 2018 accumulated or is depleted gradually over the time, subject to a random environment constructed by the Available online 6 September 2018 DRX mechanism. Therefore, the DRX mechanism acts as the background process for the fluid queue model MSC: of the battery charge. In a UE, battery charge available at any time t is considered as the fluid available 00-01 in the reservoir. Different discharging rates are considered for the various states of the DRX mechanism. 99-00 Using the fluid queue model, cumulative distribution function for the battery life of a DRX enabled UE is Keywords: derived. Further, the expression for the mean battery life is obtained and analyzed numerically. -
A Bitcoin-Inspired Infinite-Server Model with a Random Fluid Limit
A BITCOIN-INSPIRED INFINITE-SERVER MODEL WITH A RANDOM FLUID LIMIT MARIA FROLKOVA & MICHEL MANDJES Abstract. The synchronization process inherent to the Bitcoin network gives rise to an infinite- server model with the unusual feature that customers interact. Among the closed-form character- istics that we derive for this model is the busy period distribution which, counterintuitively, does not depend on the arrival rate. We explain this by exploiting the equivalence between two specific service disciplines, which is also used to derive the model's stationary distribution. Next to these closed-form results, the second major contribution concerns an asymptotic result: a fluid limit in the presence of service delays. Since fluid limits arise under scalings of the law-of-large-numbers type, they are usually deterministic, but in the setting of the model discussed in this paper the fluid limit is random (more specifically, of growth-collapse type). Key words. Stochastic-process limit ◦ fluid limit ◦ growth-collapse process ◦ Bitcoin. Affiliations. M. Frolkova and M. Mandjes are with Korteweg-de Vries Institute for Mathemat- ics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, the Netherlands. M. Frolkova is corresponding author (email: [email protected]). 1. Introduction The Bitcoin network is a payment system where all transactions between participants are carried out in the digital Bitcoin currency. These transactions are stored in a database called blockchain. In fact, every Bitcoin user maintains a blockchain version, thus keeping track of the global Bitcoin transaction history. This ensures transparency | one of the advertized Bitcoin features. Informa- tion spreads across the network via peer-to-peer communication. -
Optimal Control of a Time-Varying Double-Ended Production Queueing Model
Stochastic Systems APRIL 8, 2019 OPTIMAL CONTROL OF A TIME-VARYING DOUBLE-ENDED PRODUCTION QUEUEING MODEL By Chihoon Lee∗, Xin Liuy, Yunan Liuz, and Ling Zhangz, Stevens Institute of Technology∗,Clemson Universityy, and North Carolina State University z Motivated by production systems with nonstationary stochastic demand, we study a double-ended queueing model having backorders and customer abandonment. One side of our model stores backorders, and the other side represents inventory. We assume first-come-first- serve instantaneous fulfillment discipline. Our goal is to determine the optimal (nonstationary) production rate over a finite time hori- zon to minimize the costs incurred by the system. In addition to the inventory-related (holding, perishment) and demand-related (wait- ing, abandonment) costs, we consider a cost that penalizes rapid fluctuations of production rates. We develop a deterministic fluid control problem (FCP) that serves as a performance lower bound for the original queueing control problem (QCP). We further consider a high-volume system and construct an asymptotically optimal produc- tion rate for the QCP, under which the FCP lower bound is achieved asymptotically. Demonstrated by numerical examples, the proposed asymptotically optimal production rate successfully captures the time variability of the nonstationary demand. 1. Introduction. One of the fundamental challenges that precludes de- sired dynamic behavior of stochastic systems is that of nonstationarity. For instance, a number of service systems, such as production lines, call centers, hospitals, and online trading, are subject to nonstationarity of customer demand, service times, and staffing levels. In a pharmaceutical industry context, factors such as consumer perceptions (e.g., from advertising, time- since-entry), over-the-counter offerings, on-going market competition may lead to nonstationary demand behavior over time; for instance, Figure 1 in [3] shows monthly prescriptions fluctuation for brand and generic drugs. -
3 Lecture 3: Queueing Networks and Their Fluid and Diffusion Approximation
3 Lecture 3: Queueing Networks and their Fluid and Diffusion Approximation • Fluid and diffusion scales • Queueing networks • Fluid and diffusion approximation of Generalized Jackson networks. • Reflected Brownian Motions • Routing in single class queueing networks. • Multi class queueing networks • Stability of policies for MCQN via fluid • Optimization of MCQN — MDP and the fluid model. • Formulation of the fluid problem for a MCQN 3.1 Fluid and diffusion scales As we saw, queues can only be studied under some very detailed assumptions. If we are willing to assume Poisson arrivals and exponetial service we get a very complete description of the queuing process, in which we can study almost every phenomenon of interest by analytic means. However, much of what we see in M/M/1 simply does not describe what goes on in non M/M/1 systems. If we give up the assumption of exponential service time, then it needs to be replaced by a distribution function G. Fortunately, M/G/1 can also be analyzed in moderate detail, and quite a lot can be said about it which does not depend too much on the detailed form of G. In particular, for the M/G/1 system in steady state, we can calculate moments of queue length and waiting times in terms of the same moments plus one of the distribution G, and so we can evaluate performance of M/G/1 without too many details on G, sometimes 2 E(G) and cG suffice. Again, behavior of M/G/1 does not describe what goes on in GI/G/1 systems. -
FLUID and DIFFUSION LIMITS for QUEUES in SLOWLY CHANGING ENVIRONMENTS G. L. CHOUDHURY A. MANDELBAUM AT&T Laboratories Facult
FLUID AND DIFFUSION LIMITS FOR QUEUES IN SLOWLY CHANGING ENVIRONMENTS G. L. CHOUDHURY A. MANDELBAUM AT&T Laboratories Faculty of Industrial and Room 1L-238 Management Engineering Holmdel, NJ 07730-3030 The Technion, Haifa 32000 ISRAEL M. I. REIMAN W. WHITT Bell Labs AT&T Laboratories Lucent Technologies Room 2C-178 Room 2C-117 Murray Hill, NJ 07974-0636 Murray Hill, NJ 07974-0636 Stochastic Models 13 (1997) 121{146 (Received the first Marcel F. Neuts best paper award) Key Words: queues in random environments, nearly-completely-decomposable Markov chains, Markov-modulated process, piecewise-stationary Mt=Gt=1 queue, stochastic fluid models, diffusion processes, fluid limit, heavy traffic. ABSTRACT We consider an infinite-capacity s-server queue in a finite-state random envi- ronment, where the traffic intensity exceeds 1 in some environment states and the environment states change slowly relative to arrivals and service comple- tions. Queues grow in unstable environment states, so that it is useful to look at the system in the time scale of mean environment-state sojourn times. As the mean environment-state sojourn times grow, the queue-length and work- load processes grow. However, with appropriate normalizations, these pro- cesses converge to fluid processes and diffusion processes. The diffusion process in a random environment is a refinement of the fluid process in a random envi- ronment. We show how the scaling in these limits can help explain numerical results for queues in slowly changing random environments. For that purpose, we apply recently developed numerical-transform-inversion algorithms for the MAP/G/1 queue and the piecewise-stationary Mt=Gt=1 queue.