Queueing Theory
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Computer Systems Modelling Ian Leslie Notes by Dr R.J. Gibbens (minor edits by Ian Leslie) Computer Laboratory University of Cambridge Computer Science Tripos, Part II Lent Term 2016/17 CSM 2016/17 (1) Course overview 12 lectures covering: ◮ Introduction to modelling: ◮ what is it and why is it useful? ◮ Simulation techniques: ◮ random number generation, ◮ Monte Carlo simulation techniques, ◮ statistical analysis of results from simulation and measurements; ◮ Queueing theory: ◮ applications of Markov Chains, ◮ single/multiple servers, ◮ queues with finite/infinite buffers, ◮ queueing networks. CSM 2016/17 (2) Recommended books Ross, S.M. Probability Models for Computer Science Academic Press, 2002 Mitzenmacher, M & Upfal, E. Probability and computing: randomized algorithms and probabilistic analysis Cambridge University Press, 2005 Jain, A.R. The Art of Computer Systems Performance Analysis Wiley, 1991 Kleinrock, L. Queueing Systems — Volume 1: Theory Wiley, 1975 CSM 2016/17 (3) Introduction to modelling CSM 2016/17 (4) Why model? ◮ A manufacturer may have a range of compatible systems with different characteristics — which configuration would be best for a particular application? ◮ A system is performing poorly — what should be done to improve it? Which problems should be tackled first? ◮ Fundamental design decisions may affect performance of a system. A model can be used as part of the design process to avoid bad decisions and to help quantify a cost/benefit analysis. CSM 2016/17 (5) A simple queueing system arrival queue server departure ◮ -
The Queueing Network Analyzer
THE BELL SYSTEM TECHNICAL JOURNAL Vol. 62, No.9, November 1983 Printed in U.S.A. The Queueing Network Analyzer By W. WHITT* (Manuscript received March 11, 1983) This paper describes the Queueing Network Analyzer (QNA), a software package developed at Bell Laboratories to calculate approximate congestion measures for a network of queues. The first version of QNA analyzes open networks of multiserver nodes with the first-come, first-served discipline and no capacity constraints. An important feature is that the external arrival processes need not be Poisson and the service-time distributions need not be exponential. Treating other kinds of variability is important. For example, with packet-switchedcommunication networks we need to describe the conges tion resulting from bursty traffic and the nearly constant service times of packets. The general approach in QNA is to approximately characterize the arrival processes by two or three parameters and then analyze the individual nodes separately. The first version of QNA uses two parameters to characterize the arrival processes and service times, one to describe the rate and the other to describe the variability. The nodes are then analyzed as standard GI/G/m queues partially characterized by the first two moments of the interarrival time and service-time distributions. Congestion measures for the network as a whole are obtained by assuming as an approximation that the nodes are stochastically independent given the approximate flow parameters. I. INTRODUCTION AND SUMMARY Networks of queues have proven to be useful models to analyze the performance of complex systems such as computers, switching ma chines, communications networks, and production job shOpS.1-7 To facilitate the analysis of these models, several software packages have * Bell Laboratories. -
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. -
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. -
A Simple, Practical Prioritization Scheme for a Job Shop Processing Multiple Job Types
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-2013 A Simple, Practical Prioritization Scheme for a Job Shop Processing Multiple Job Types Shuping Zhang [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Management Sciences and Quantitative Methods Commons Recommended Citation Zhang, Shuping, "A Simple, Practical Prioritization Scheme for a Job Shop Processing Multiple Job Types. " PhD diss., University of Tennessee, 2013. https://trace.tennessee.edu/utk_graddiss/2503 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Shuping Zhang entitled "A Simple, Practical Prioritization Scheme for a Job Shop Processing Multiple Job Types." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Management Science. Mandyam M. Srinivasan, Melissa Bowers, Major Professor We have read this dissertation and recommend its acceptance: Ken Gilbert, Theodore P. Stank Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) A Simple, Practical Prioritization Scheme for a Job Shop Processing Multiple Job Types A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Shuping Zhang August 2013 Copyright © 2013 by Shuping Zhang All rights reserved. -
(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. -
Introduction to Queueing Theory Review on Poisson Process
Contents ELL 785–Computer Communication Networks Motivations Lecture 3 Discrete-time Markov processes Introduction to Queueing theory Review on Poisson process Continuous-time Markov processes Queueing systems 3-1 3-2 Circuit switching networks - I Circuit switching networks - II Traffic fluctuates as calls initiated & terminated Fluctuation in Trunk Occupancy Telephone calls come and go • Number of busy trunks People activity follow patterns: Mid-morning & mid-afternoon at All trunks busy, new call requests blocked • office, Evening at home, Summer vacation, etc. Outlier Days are extra busy (Mother’s Day, Christmas, ...), • disasters & other events cause surges in traffic Providing resources so Call requests always met is too expensive 1 active • Call requests met most of the time cost-effective 2 active • 3 active Switches concentrate traffic onto shared trunks: blocking of requests 4 active active will occur from time to time 5 active Trunk number Trunk 6 active active 7 active active Many Fewer lines trunks – minimize the number of trunks subject to a blocking probability 3-3 3-4 Packet switching networks - I Packet switching networks - II Statistical multiplexing Fluctuations in Packets in the System Dedicated lines involve not waiting for other users, but lines are • used inefficiently when user traffic is bursty (a) Dedicated lines A1 A2 Shared lines concentrate packets into shared line; packets buffered • (delayed) when line is not immediately available B1 B2 C1 C2 (a) Dedicated lines A1 A2 B1 B2 (b) Shared line A1 C1 B1 A2 B2 C2 C1 C2 A (b) -
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. -
Queueing Theory
Queueing theory Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory in 1909.[8][9][10] He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920.[11] In Kendall’s notation: • M stands for Markov or memoryless and means ar- rivals occur according to a Poisson process • D stands for deterministic and means jobs arriving at the queue require a fixed amount of service • k describes the number of servers at the queueing node (k = 1, 2,...). If there are more jobs at the node than there are servers then jobs will queue and wait for service Queue networks are systems in which single queues are connected The M/M/1 queue is a simple model where a single server by a routing network. In this image servers are represented by serves jobs that arrive according to a Poisson process and circles, queues by a series of retangles and the routing network have exponentially distributed service requirements. In by arrows. In the study of queue networks one typically tries to an M/G/1 queue the G stands for general and indicates obtain the equilibrium distribution of the network, although in an arbitrary probability distribution. The M/G/1 model many applications the study of the transient state is fundamental. was solved by Felix Pollaczek in 1930,[12] a solution later recast in probabilistic terms by Aleksandr Khinchin and Queueing theory is the mathematical study of waiting now known as the Pollaczek–Khinchine formula.[11] lines, or queues.[1] In queueing theory a model is con- structed so that queue lengths and waiting time can be After World War II queueing theory became an area of [11] predicted.[1] Queueing theory is generally considered a research interest to mathematicians. -
Delay Models in Data Networks
3 Delay Models in Data Networks 3.1 INTRODUCTION One of the most important perfonnance measures of a data network is the average delay required to deliver a packet from origin to destination. Furthennore, delay considerations strongly influence the choice and perfonnance of network algorithms, such as routing and flow control. For these reasons, it is important to understand the nature and mechanism of delay, and the manner in which it depends on the characteristics of the network. Queueing theory is the primary methodological framework for analyzing network delay. Its use often requires simplifying assumptions since, unfortunately, more real- istic assumptions make meaningful analysis extremely difficult. For this reason, it is sometimes impossible to obtain accurate quantitative delay predictions on the basis of queueing models. Nevertheless, these models often provide a basis for adequate delay approximations, as well as valuable qualitative results and worthwhile insights. In what follows, we will mostly focus on packet delay within the communication subnet (i.e., the network layer). This delay is the sum of delays on each subnet link traversed by the packet. Each link delay in tum consists of four components. 149 150 Delay Models in Data Networks Chap. 3 1. The processinR delay between the time the packet is correctly received at the head node of the link and the time the packet is assigned to an outgoing link queue for transmission. (In some systems, we must add to this delay some additional processing time at the DLC and physical layers.) 2. The queueinR delay between the time the packet is assigned to a queue for trans- mission and the time it starts being transmitted. -
Matrix Geometric Approach for Random Walks
Matrix geometric approach for random walks Citation for published version (APA): Kapodistria, S., & Palmowski, Z. B. (2017). Matrix geometric approach for random walks: stability condition and equilibrium distribution. Stochastic Models, 33(4), 572-597. https://doi.org/10.1080/15326349.2017.1359096 Document license: CC BY-NC-ND DOI: 10.1080/15326349.2017.1359096 Document status and date: Published: 02/10/2017 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.