Markov Chains and Queues

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50 INTRODUCTION

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Markov chain theory has numerous applications to queueing systems This chapter gives a rst introduction to the analysis of queues and stochastic networks In Section 51 we consider the Erlang delay model with Poisson arrivals and exponential services We rst analyse the single-server M/M/1 queue and next the multi-server M/M/c queue Section 52 deals with both the Erlang loss model with Poisson input and the Engset loss model with nite-source input The Erlang delay model and Erlang s loss formula will be used in Section 53 to obtain a square-root staf ng rule for the design of stochastic service systems The Erlang loss model and the Engset loss model have the so-called insensitivity property stating that the equilibrium distribution of the number of customers present is insensitive to the form of the service-time distribution and requires only the mean service time This insensitivity property, being of utmost importance in practice, will be discussed in a more general framework in Section 54 The so-called phase method is the subject of Section 55 This powerful method uses the idea that any probability distribution function of a non-negative random variable can be arbitrarily closely approximated by a mixture of Erlangian distributions with the same scale parameters This fundamental result greatly enhances the applicability of the continuous-time Markov chain model In Section 56 the theory of continuous-time Markov chains will be used to analyse open and closed queueing networks In particular, a product-form formula will be established for the joint distribution of the number of customers present at the various nodes of the network

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THE ERLANG DELAY MODEL

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Consider a multi-server station at which customers arrive according to a Poisson process with rate There are c servers with a shared in nite-capacity waiting line If an arriving customer nds a free server, the customer immediately enters service; otherwise, the customer joins the queue The service times of the customers are

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A First Course in Stochastic Models HC Tijms c 2003 John Wiley & Sons, Ltd ISBNs: 0-471-49880-7 (HB); 0-471-49881-5 (PB)

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MARKOV CHAINS AND QUEUES

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independent random variables having a common exponential distribution with mean 1/ The service times and the arrival process are independent of each other Let = c (511)

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It is assumed that < 1 Rewriting this condition as / < c, the condition states that the average amount of work offered to the servers per time unit is less than the total service capacity The factor is called the server utilization This model is a basic model in queueing theory and is often called the Erlang delay model It is usually abbreviated as the M/M/c queue Using continuous-time Markov chain theory we will derive the distributions of the queue size and the delay in queue of a customer Let X(t) = the number of customers present at time t (including any customer in service) Then the stochastic process {X(t)} is a continuous-time Markov chain with in nite state space I = {0, 1, } The assumption < 1 implies that the Markov chain satis es Assumption 421 with regeneration state 0 and thus has a unique equilibrium distribution {pj } (a formal proof is omitted) The probability pj gives the long-run fraction of time that j customers are present 511 The M/M/1 Queue For ease of presentation, we rst analyse the single-server case with c = 1 The transition rate diagram of the process {X(t)} is given in Figure 511 Note that for each state i the transition rate qij = 0 for j i 2 This implies that the equilibrium probabilities pj can be recursively computed; see formula (4210) By equating the rate at which the process leaves the set {i, i + 1, } to the rate at which the process enters this set, it follows that pi = pi 1 , i = 1, 2,

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The recurrence equation allows for an explicit solution Iterating the equation yields pi = ( / )i p0 for all i 1 Noting that this relation also holds for i = 0 and substituting it into the normalizing equation pi = 1, we nd p0 (1 / ) 1 = i=0

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