Approximating Heavy Traffic with Brownian Motion
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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. -
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. -
Queueing-Theoretic Solution Methods for Models of Parallel and Distributed Systems·
1 Queueing-Theoretic Solution Methods for Models of Parallel and Distributed Systems· Onno Boxmat, Ger Koolet & Zhen Liui t CW/, Amsterdam, The Netherlands t INRIA-Sophia Antipolis, France This paper aims to give an overview of solution methods for the performance analysis of parallel and distributed systems. After a brief review of some important general solution methods, we discuss key models of parallel and distributed systems, and optimization issues, from the viewpoint of solution methodology. 1 INTRODUCTION The purpose of this paper is to present a survey of queueing theoretic methods for the quantitative modeling and analysis of parallel and distributed systems. We discuss a number of queueing models that can be viewed as key models for the performance analysis and optimization of parallel and distributed systems. Most of these models are very simple, but display an essential feature of dis tributed processing. In their simplest form they allow an exact analysis. We explore the possibilities and limitations of existing solution methods for these key models, with the purpose of obtaining insight into the potential of these solution methods for more realistic complex quantitative models. As far as references is concerned, we have restricted ourselves in the text mainly to key references that make a methodological contribution, and to sur veys that give the reader further access to the literature; we apologize for any inadvertent omissions. The reader is referred to Gelenbe's book [65] for a general introduction to the area of multiprocessor performance modeling and analysis. Stochastic Petri nets provide another formalism for modeling and perfor mance analysis of discrete event systems. -
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 -
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. -
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) -
Analysis of Packet Queueing in Telecommunication Networks
Analysis of Packet Queueing in Telecommunication Networks Course Notes. (BSc.) Network Engineering, 2nd year Video Server TV Broadcasting Server AN6 Web Server File Storage and App Servers WSN3 R1 AN5 AN7 AN4 User9 WSN1 R2 R3 User8 AN1 User7 User1 R4 User2 AN3 User6 User3 WSN2 AN2 User5 User4 Boris Bellalta and Simon Oechsner 1 . Copyright (C) by Boris Bellalta and Simon Oechsner. All Rights Reserved. Work in progress! Contents 1 Introduction 8 1.1 Motivation . .8 1.2 Recommended Books . .9 I Preliminaries 11 2 Random Phenomena in Packet Networks 12 2.1 Introduction . 12 2.2 Characterizing a random variable . 13 2.2.1 Histogram . 14 2.2.2 Expected Value . 16 2.2.3 Variance . 16 2.2.4 Coefficient of Variation . 17 2.2.5 Moments . 17 2.3 Stochastic Processes with Independent and Dependent Outcomes . 18 2.4 Examples . 19 2.5 Formulation of independent and dependent processes . 23 3 Markov Chains 25 3.1 Introduction and Basic Properties . 25 3.2 Discrete Time Markov Chains: DTMC . 28 3.2.1 Equilibrium Distribution . 28 3.3 Continuous Time Markov Chains . 30 3.3.1 The exponential distribution in CTMCs . 31 3.3.2 Memoryless property of the Exponential Distribution . 33 3.3.3 Equilibrium Distribution . 34 3.4 Examples . 35 3.4.1 Load Balancing in a Farm Server . 35 3.4.2 Performance Analysis of a Video Server . 36 2 CONTENTS 3 II Modeling the Internet 40 4 Delays in Communication Networks 41 4.1 A network of queues . 41 4.2 Types of Delay . -
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. -
Telephone Call Centers: Tutorial, Review, and Research Prospects
University of Pennsylvania ScholarlyCommons Operations, Information and Decisions Papers Wharton Faculty Research 2003 Telephone Call Centers: Tutorial, Review, and Research Prospects Noah Gans University of Pennsylvania Ger Koole Avishai Mandelbaum Follow this and additional works at: https://repository.upenn.edu/oid_papers Part of the Business Administration, Management, and Operations Commons, Growth and Development Commons, Human Resources Management Commons, and the Operations and Supply Chain Management Commons Recommended Citation Gans, N., Koole, G., & Mandelbaum, A. (2003). Telephone Call Centers: Tutorial, Review, and Research Prospects. manufacturing & Service Operations Management, 5 (2), 79-141. http://dx.doi.org/10.1287/ msom.5.2.79.16071 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/oid_papers/136 For more information, please contact [email protected]. Telephone Call Centers: Tutorial, Review, and Research Prospects Abstract Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating sociotechnical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments traditional operational models are of great value—and at the same time fundamentally limited—in their ability to characterize system performance. We review the state of research on telephone call centers. We begin with a tutorial on how call centers function and proceed to survey -
Queueing Systems
Queueing Systems Ivo Adan and Jacques Resing Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands March 26, 2015 Contents 1 Introduction7 1.1 Examples....................................7 2 Basic concepts from probability theory 11 2.1 Random variable................................ 11 2.2 Generating function............................... 11 2.3 Laplace-Stieltjes transform........................... 12 2.4 Useful probability distributions........................ 12 2.4.1 Geometric distribution......................... 12 2.4.2 Poisson distribution........................... 13 2.4.3 Exponential distribution........................ 13 2.4.4 Erlang distribution........................... 14 2.4.5 Hyperexponential distribution..................... 15 2.4.6 Phase-type distribution......................... 16 2.5 Fitting distributions.............................. 17 2.6 Poisson process................................. 18 2.7 Exercises..................................... 20 3 Queueing models and some fundamental relations 23 3.1 Queueing models and Kendall's notation................... 23 3.2 Occupation rate................................. 25 3.3 Performance measures............................. 25 3.4 Little's law................................... 26 3.5 PASTA property................................ 27 3.6 Exercises..................................... 28 4 M=M=1 queue 29 4.1 Time-dependent behaviour........................... 29 4.2 Limiting behavior............................... -
A Survey on Performance Analysis of Warehouse Carousel Systems
A Survey on Performance Analysis of Warehouse Carousel Systems N. Litvak∗, M. Vlasiou∗∗ August 6, 2018 ∗ Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands. ∗∗ Eurandom and Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected], [email protected] Abstract This paper gives an overview of recent research on the performance evaluation and design of carousel systems. We discuss picking strategies for problems involving one carousel, consider the throughput of the system for problems involving two carousels, give an overview of related problems in this area, and present an extensive literature review. Emphasis has been given on future research directions in this area. Keywords: order picking, carousels systems, travel time, throughput AMS Subject Classification: 90B05, 90B15 1 Introduction A carousel is an automated storage and retrieval system, widely used in modern warehouses. It consists of a number of shelves or drawers, which are linked together and are rotating in a closed loop. It is operated by a picker (human or robotic) that has a fixed position in front of the carousel. A typical vertical carousel is given in Figure1. arXiv:1404.5591v1 [math.PR] 22 Apr 2014 Carousels are widely used for storage and retrieval of small and medium-sized items, such as health and beauty products, repair parts of boilers for space heating, parts of vacuum cleaners and sewing machines, books, shoes and many other goods. In e-commerce companies use carousel to store small items and manage small individual orders.