ALGORITHMIC ANALYSIS OF QUEUEING MODELS

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the M/D/c queue will be given in Section 961 In Section 962 we consider the M/G/c queue with general service times and give several approximations including two-moment approximations based on exact results for the M/M/c queue and the M/D/c queue In Section 963 we consider the M X /G/c queue with batch arrivals and general service times In particular, the M X /M/c queue and the M X /D/c queue are dealt with 961 The M/D/c Queue In this model the arrival process of customers is a Poisson process with rate , the service time of a customer is a constant D, and c identical servers are available It is assumed that the server utilization = D/c is smaller than 1 An exact algorithm analysis of the M/D/c queue goes back to Crommelin (1932) and is based on the following observation Since the service times are equal to the constant D, any customer in service at time t will have left the system at time t + D, while the customers present at time t + D are exactly those customers either waiting in queue at time t or having arrived in (t, t + D) Let pj (s) be the probability of having j customers in the system at time s Then, by conditioning on the number of customers present at time t,

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for j = 0, 1, , since the number of arrivals in a time D is Poisson distributed with mean D Next, by letting t in these equations, we nd that the time-average probabilities pj satisfy the linear equations pj = e D ( D)j j!

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(961)

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Also, we have the normalizing equation j =0 pj = 1 This in nite system of linear equations can be reduced to a nite system of linear equations by using the geometric tail approach discussed in Section 342 It will be shown below that the state probabilities pj exhibit the geometric tail behaviour

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where is the unique solution of the equation e D(1 ) c = 1 on the interval (1, ) and the constant is given by = (c D ) 1

c 1 k=0

(963)

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(964)

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Since pj /pj 1 1 for j large enough, we replace pj for j M by pM (j M) for an appropriately chosen integer M Then the in nite system of linear equations (961) together with the normalizing equation j =0 pj = 1 is reduced to a nite system of linear equations of dimension M + 1 A relatively small value of M is usually good enough for practical purposes The value of M does not grow beyond any practical bound when the traf c load on the system gets close to 1 It is an empirical fact that the asymptotic expansion (962) already applies for relatively small values of j For practical purposes the value M = 1 (1 + )c + 10 c seems 2 large enough to obtain the state probabilities to at least nine decimal places (eg for c = 25 and = 099 we have M = 75, which is in marked contrast with the brute-force value N = 1056 that is required when the in nite system of linear equations is truncated such that i=N pi 10 9 ) In general the geometric tail approach leads to a relatively small system of linear equations that can usually be solved by a standard Gaussian elimination method This approach requires that beforehand we compute the constant from (963) Using logarithms, the equation (963) is equivalent to D(1 ) + c ln( ) = 0 Noting that D = c and using the transformation = 1/ , it follows that can be obtained by computing the unique (0, 1) satisfying (1 ) + ln( ) = 0 We can conclude that the state probabilities in the M/D/c queue can be routinely computed by solving a nite system of linear equations An accuracy check on the calculated values of the pj is Little s relation