Tai Lieu Chat Luong An Elementary Introduction to QUEUEING SYSTEMS 9190_9789814612005_tp.indd 5/5/14 3:20 pm May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory This page intentionally left blank PST˙ws An Elementary Introduction to QUEUEING SYSTEMS Wah Chun Chan University of Calgary, Canada World Scientific NEW JERSEY • LONDON 9190_9789814612005_tp.indd • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 5/5/14 3:20 pm Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library AN ELEMENTARY INTRODUCTION TO QUEUEING SYSTEMS Copyright © 2014 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 978-981-4612-00-5 Printed in Singapore This book is dedicated to the memory of my uncle and aunt, Mr and Mrs Lap Hoi Chan, who supported me during my youth, and my professor, Dr Donald A George, who inspired me in the study of the theory of probability v May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory This page intentionally left blank PST˙ws ACKNOWLEDGEMENTS The author wishes to thank his wife, Yu-Chih, and his family members, Eileen and Al, Jean and Aaron, Vivian and Brian, and An-Wen for their encouragement and support during the preparation of the book Also, a special thanks to Eileen for her skillful typing of the manuscript in her busy work schedule vii May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory This page intentionally left blank PST˙ws CONTENTS Preface Chapter Modeling of Queueing Systems 1.1 Mathematical Modeling 1.2 The Poisson Input Process 1.3 Superposition of Independent Poisson Processes 1.4 Decomposition of a Poisson Process 10 1.5 The Exponential Interarrival Time Distribution 12 1.6 The Markov Property or Memoryless Property 13 1.7 Relationship Between the Poisson Distribution and the Exponential Distribution 14 1.8 The Service Time Distribution 15 1.9 The Residual Service Time Distribution 17 1.10 The Birth and Death Process 19 1.11 The Outside Observer’s Distribution and the Arriving Customer’s Distribution 25 Chapter Queueing Systems with Losses 29 2.1 Introduction 29 2.2 The Erlang Loss System 30 2.3 The Erlang Loss Formula 31 Chapter Queueing Systems Allowing Waiting 41 3.1 Introduction 41 3.2 The Erlang Delay System 41 ix 89 Queueing Systems with A Single Server = W0j* j-1 j * + ∑ ρi Wi + ∑ ρi Wj* i=1 i=1 Solving the equation for Wj* yields j-1 Wj* = W0j* + ∑ ρi Wi* i=1 j – ∑ ρi i=1 , j = 1, 2, m From this expression, we find Wj* = W0j* , j = 1, 2, , m (1 – δj) (1 – δj-1) (5.21) j where δj = ∑ ρi, δ0 = 0, and W0j* is given by (5.20), which is the i=1 average waiting time of customers in the preemptive priority system on their first arrival However, extra waiting time is introduced due to the fact that a customer of class j can be preempted (interrupted) by higher priority customers during service with probability δj-1 , on average Thus, the extra delay due to the service of a class j customer is δj-1 X j Furthermore, this extra delay will introduce an additional delay equal to δj-1 (δj-1 X j ) Continuing this reasoning reveals that the extra average delay of a class j customer to complete his service is (δj-1 + δ2j-1 + ) Xj = δj-1 Xj – δj-1 90 An Elementary Introduction to Queueing Systems Finally, the average waiting time for a class j customer is the sum of Wj* in (5.21) and this extra delay: Wj = δj-1 Xj – δj-1 + W0j* , j = 1, 2, , m (1 – δj) (1 – δj-1) (5.22) where W0j* is given by (5.20) 5.6 The GI/M/1 Queue This section will study the single server queue for which the interarrival times of customers are mutually independent and identically distributed, and the service time has an exponential distribution Suppose that customers arrive at time epochs T1, T2, We assume that the interarrival times Xk+1 = Tk+1 – Tk , k = 0, 1, ; T0 = are mutually independent and identically distributed random variables and have the general distribution function G(x) = P{Xk+1 ≤ x} , k = 0, 1, and mean 1/λ The service time has an exponential distribution with mean 1/µ Let Nk denote the number of customers in the system just prior to the arrival of the kth customer, that is, Nk denotes the number of customers present at time Tk – Fig 5-5 depicts the (k+1)th interarrival time Xk+1 and the Poisson departure process (see Section 1.7) 91 Queueing Systems with A Single Server kth arrival (k+1)th arrival Xk+1 _ time Tk Tk+1 Nk, πi Poisson departures with rate µ Nk+1, πj Fig 5-5 The kth interarrival time and the departure process Applying the formula of total probability, we can write the state probability ∞ P{Nk+1 = j} = ∑ P{Nk+1 = j | Nk = i} P{Nk = i} i=0 Since the conditional probability depends on the indices i and j but not on the index k, we let pij = P{Nk+1 = j | Nk = i} Under the equilibrium condition ρ = λ/ µ < 1, an unique stationary state probability distribution πj = lim P{Nk = j} k∞ exists It follows from the formula of total probability that as k∞, we have, in the steady state 92 An Elementary Introduction to Queueing Systems ∞ πj = ∑ pij πi , j = 0, 1, … i=0 which is subject to the normalization condition ∞ ∑ πj = j=0 We now proceed to determine the transition probability pij First, observe that if Nk = i and Nk+1 = j, then the value of j cannot be greater than i+1 It follows that pij = 0, j > i+1 For j ≤ i+1, note that the server has been busy continuously during the interarrival time Xk+1, the departure process obeys the Poisson distribution with rate µ because the service times are exponentially distributed with mean 1/ µ It follows that the probability of exactly i–1+j departures during a given time interval of length x is P{Nk+1 = j | Nk = i, Xk+1 = x} = (µ x)i+1-j e-µ x (i+1-j)! Since Xk+1 has the distribution function G(x), then the transition probability pij is given by ∞ pij = ∫ (µ x)i+1-j e-µ x dG(x) , i ≥ j – , j = 0, 1, … (i+1– j)! It follows that ∞ ∞ πj = ∑ πi ∫ (µ x)i+1-j e-µ x dG(x) , i ≥ j – i=j-1 (i + 1– j)! Queueing Systems with A Single Server ∞ 93 ∞ = ∑ πk+j-1 ∫ (µ x)k e-µ x dG(x) , j = 0, 1, k=0 k! and the state probability distribution is now completely determined Recall that the state probability πj is defined as the probability that the system is in state j right before the arrival epoch of customers In fact, the arrival epochs here are regeneration points To find πj , we assume a solution of the form πj = A δj , j = 0, 1, … , < δ < where A and δ are unknown quantities and remain to be determined Using this solution in the last expression yields ∞ δ = ∫ e-(1 - δ) µ x dG(x) = Ĝ((1 – δ) µ) (5.23) where Ĝ((1 – δ) µ) is the Laplace –Stieljes transform of the function G(x) The function Ĝ((1 – δ) µ) has the following properties: ∞ Ĝ(µ) = Ĝ((1 – δ) µ) | δ=0 = ∫ e-µ x dG(x) > 0 and Ĝ(0) = Ĝ((1 – δ) µ) | = Ĝ(0) = δ=1 Further, we also have dĜ dδ = µ = >1 λ ρ δ=1 Thus, an unique real solution δ0 exists for δ in the range < δ < This is shown in Fig 5-6 94 An Elementary Introduction to Queueing Systems Ĝ((1 – δ) µ) Ĝ(µ) δ δ0 Fig 5-6 Real solution of Ĝ((1 – δ) µ) = δ It remains to determine the constant A Using the normalization condition gives ∞ ∞ ∑ πj = ∑ A δj = j=0 j=0 Hence, A=1–δ and the state probability distribution is then given by πj = (1 – δ) δj , j = 0, 1, … (5.24) where δ is given by the real solution δ0 of (5.23) (A) The Probability of Waiting and the Mean Waiting Time The other quantities of practical interest are the probability of waiting and the mean waiting time We let W denote the waiting 95 Queueing Systems with A Single Server time If an arriving customer finds the server busy, he must wait Thus, the probability of waiting is simply pw = P{W > 0} = – π0 = δ (5.25) Using exactly the same reasoning as in the derivation of the Pollaczek-Khinchin formula for the mean waiting time, except that the probability of waiting is δ now instead of ρ and the E[R] is simply the mean service time τ because of the exponential service time, we immediately obtain the mean waiting time for the GI/M/1 queue E[W] = δ E[R] = δ τ 1–δ 1–δ (5.26) (B) The Waiting Time Distribution Function Following exactly the same reasoning as the investigation of the waiting time distribution function for the M/M/1 queue, we write ∞ P {W > t} = ∑ πk Pk {W > t} k=1 ∞ ∞ = ∑ (1 – δ) δk ∫ µk xk-1 e-µ x dx t (k – 1)! k=1 = δ e-µ (1 - δ) t (5.27) where P0{W > t} = and δ is the unique solution δ0 of the equation δ = Ĝ ((1 – δ)µ) in the range < δ < Comments Similar to the M/G/1 queue case, for the GI/M/1 queue, if the interarrival time distribution is exponential with mean 1/λ, then the GI/M/1 queue becomes the M/M/1 queue All results obtained for the GI/M/1 queue should become those for the M/M/1 96 An Elementary Introduction to Queueing Systems queue Now let us assume that the interarrival time distribution function is given by G(x) = – e-λ x Then the Laplace-Stieljes transform of G(x) in (5.22) becomes ∞ Ĝ[(1 – δ) µ] = ∫ e-µ ( – δ )x dG(x) ∞ = ∫ e-µ ( – δ )x λ e-λ x dx = λ µ ( – δ) + λ And (5.23) becomes δ= λ µ ( – δ) + λ Hence, we find δ= λ µ =ρ which is the real solution of (5.23) The state probability distribution in (5.24) is then given by πj = ( – ρ) ρj , j = 0, 1, which is the same as pk in (5.1) for the M/M/1 queue Other results of (5.25) to (5.27) follow immediately Queueing Systems with A Single Server 97 Review The simplest queueing system is the one with Poisson input and a single exponential server The formulas for calculating the performance indices are very simple When the service time distribution is general, only the Pollaczek-Khinchin formula for the mean waiting time is available We have presented a direct derivation of this formula without using the concept of imbedded Markov chain Furthermore, formulas for the mean waiting time are extended to the case of server vacations and the case of priority discipline For the system with a general input process and a single exponential server, formulas of the probability of waiting, the mean waiting time and the waiting time distribution function are obtained These closed form formulas are expressed in terms of a real solution of a functional equation (5.23), which is related to the distribution function of the general input process If the general input process is a Poisson input process, then the real solution will become the offered load to the system May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory This page intentionally left blank PST˙ws BIBLIOGRAPHY [1] Kendall, D.G., “Some Problems in the Theory of Queues”, J Roy, Statist Soc (B), 1951, Vol 13, 151 – 185 [2] Feller, W., An Introduction to Probability Theory and its Applications, Vol 1, 3rd edn., Wiley, 1968 [3] Khinchin, A.Y., Mathematical Methods in the Theory of Queueing, 2nd edn., Hafner Publishing Co., 1969 [4] Kleinrock, L., Queueing Systems: Vol - Theory, Wiley, 1976 [5] Cooper, R.B., Introduction to Queueing Theory, 2nd edn., Elsevier Science, 1981 [6] Erlang, A.K., “Solution of Some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges”, The Post Office Electrical Engineers J., 1917, Vol 10, 189 – 197 [7] Pinsky, E., Conway, A and Liu, W., “Blocking Formulae for the Engset Model”, IEEE Trans Communications, 1994, Com – 42, 2213 – 2214 [8] Wolft, R.W., Poisson Arrivals See Time Average, Opns Res., 1982, Vol 30, 223 - 231 [9] Chan, W.C and Lin, Y.B., “Waiting Time Distribution for the M/M/m Queue”, Proc IEE – Communications, June 2003, Vol 150, 159 – 162 [10] Fry, T.C., Probability and Its Engineering Uses, 2nd edn., Van Norstrand, 1965 99 100 An Elementary Introduction to Queueing Systems [11] Chan, W.C., Performance Analysis of Telecommunications and Local Area Networks, Kluwer Academic Publishers, 2000 [12] Khinchin, A.Y., Sequences of Chance Events with After-Effects, Theory of Probability and its Applications, 1956, Vol 1, 1-15 [13] Khinchin, A.Y., On Poisson Sequences of Chance Events, Theory of Probability and Applications, 1956, Vol 1, 291-297 [14] Beckmann, P., Elementary Queueing and Telephony, 1981, Rev edn., Batavia, IL.: ABC Tele Training [15] Kleinrock, L., On the Modeling and Analysis of Computer Networks, 1993, Vol 81, No.8, Proc Of IEEE, 1179-1191 [16] Chan, W.C., Modeling of Data Networks, International J of Modeling and Simulation, 1993, Vol 13, No.4, 121-128 [17] Copenhagen Telephone Co., The Life and Works of A.K Erlang, 1948 [18] Redheffer, R.M., A Note on the Poisson Law, 1953, Vol 26, No 2, 185-188 [19] Martine, R.R., Basic Traffic Analysis, Englewood Cliffs, N.J., Prentice Hall, 1994 INDEX Carried traffic, carried load, 32, 33, 35, 36, 39, 40, 45, 46, 54, 64, 69 Cobham’s formula, 87 Coefficient of variation, 79 Congestion call, 69 time, 69 Decomposition of a Poisson process, 11 Delay system, see also Erlang delay system, Engset delay system, xii, 28, 40, 42, 44, 45, 46, 55, 57, 66, 67, 68, 69, 70 Departure criteria, 51, 90, 91, 92 Distribution function of the waiting time, 41, 45 Embedded Markov chain, see imbedded Markov chain Engset delay system, 28, 66, 67, 68, 69, 70 Engset loss formula, 64 Engset loss system, 59, 60, 61, 64, 65, 66 Erlang B curves, 35, 36, 39 Arrival epoch, 3, 4, 12, 26, 28, 48, 93 Arrival process, 9, 10, 11, 12, 15, 27, 28, 49, 51, 59 Arriving customer’s distribution, 25, 26, 27, 49, 64 Average, mean, expected value arrival rate, 4, 28, 52, 53, 68, 69 queue length, 45, 53, 75, 79 service time, 18, 53, 69, 78, 80, 82, 86, 95 waiting time, xi, 45, 53, 54, 55, 56, 57, 67, 68, 69, 71, 74, 79, 80, 81, 82, 83, 84, 85, 90, 94, 95, 97 Bernoulli random variable, Bernoulli trials, Binomial distribution, 5, 62, 65 Birth and death process, xii, 19, 20, 21, 22, 24, 25, 28, 31, 37, 40, 42, 61, 70, 71, 74 Birth rate, 20 Blocking, 33, 35, 39 Blocking probability, 33, 35, 39 Busy hour, 7, 38, 39 Busy period, 83 101 102 An Elementary Introduction to Queueing Systems Erlang B formula, loss formula, loss system, xii, 33, 37, 38, 40, 42, 57 Erlang C curves, 44, 54 Erlang C formula, 44 Erlang delay system, 28, 40, 42, 44, 45, 46, 55, 57 Erlang distribution, 2, 57 Events independent, 8, 10 Exponential service time distribution, 13, 14, 15, 16, 28, 30, 33, 35, 47, 48, 49, 71, 82, 90 First-come first-served (FCFS, FIFO), 3, 30, 41, 79, 80, 85 Formula of total probability, 20, 22, 46, 91 General service time distribution, 71 GI/M/1 queue, 95 Grade of service, 38, 39, 53, 54 Holding time, 15, 16, 32, 34, 38, 54 Imbedded Markov chain, xii, 74, 75, 80, 97 Independent events, 8, 10 Interarrival time, 2, 3, 4, 12, 13, 14, 80, 90, 91, 92, 95 Khinchin, see Pollaczek-Khinchin formula, 22, 99, 100 Laplace-Stieljes transform, 19, 96 Little’s formula, 45, 53, 68, 69, 74, 79 Loss system Engset loss system, 59, 60, 61, 64, 65, 66 Erlang loss system, xii, 37, 38, 40, 42, 57 M/D/1 queue, 79 M/G/1 queue, 17, 74, 79, 80, 81, 82, 83, 95 M/M/1 queue, 71, 82, 95, 96 M/M/m queue, see Erlang system, 48 Markov chain, see Markov property, Imbedded Markov chain, xii, 19, 74, 75, 80, 97 Markov property, memoryless property, 13, 14, 28, 30, 33, 49, 50, 56, 74 Non-preemptive priority system, 84, 88 Normalization condition, 23, 32, 43, 61, 92, 94 Occupancy, 33, 36, 43 Offered load and carried load in the Engset delay system, 68 loss system, 64 Offered load per idle source, 61 Offered traffic, offered load, 32, 34, 35, 36, 39, 40, 44, 46, 53, 54, 61, 64, 68, 69, 73, 85, 97 Outside observer’s distribution, 25, 26, 27, 49, 64 Poisson arrivals, see time average (PASTA), 27, 35, 37, 49, 75 Poisson distribution, 6, 7, 14, 15, 30, 33, 47, 92 Poisson input process, xii, 8, 34, 55, 59, 71, 97 Poisson random variable, 6, 7, Pollaczek-Khinchin formula for the mean waiting time, 79, 80, 95, 97 Preemptive priority system, 88, 89 Priority discipline, 71, 97 Probability of loss, 33, 40, 41 Quality of service, xi, 29, 41 Quasi-random input, 59, 60, 69, 70 Queueing Systems with A Single Server Queueing models characteristics, 2, 41, 59 Queueing systems GI/M/1 queue, 95 M/D/1 queue, 79 M/G/1 queue, 79, 80, 81 M/M/1 queue, 71, 82, 95, 96 priority queue, 84, 85 Random process, 1, 11, 74 Random variables, xii, 4, 8, 19, 45, 50, 55, 75, 76, 90 Regeneration point, 74, 93 Relative frequency, Residual service time distribution, 28 Service time distribution, 14, 28, 30, 33, 35, 49, 50, 59, 71, 82, 97 103 Superposition of independent Poisson input processes, 8, 10 Throughput, 70 Time congestion, 69 Truncated binomial distribution, 62, 65 Truncated Poisson distribution, 33 Utilization factor, 33, 38, 40 Vacations, 71, 83, 97 Variance of the Poisson random variable, 6, Waiting time distribution function, 48, 50, 57, 66, 67, 72, 95, 97 Waiting time distribution function for Engset delay system, 67