ALGORITHMIC ANALYSIS OF QUEUEING MODELS

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from a state in the set {k + 1, k + 2, } to a state outside this set during one cycle equals the number of upcrossings from a state outside the set {k + 1, k + 2, } to a state in this set during one cycle Thus relation (925) generalizes to

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The remainder of the proof is analogous to the proof of Theorem 921 The recursion scheme (932) is not as easy to apply as the recursion scheme (921) The reason is that the computation of the constants an is quite burdensome In general, numerical integration must be used, where each function evaluation in the integration procedure requires an application of Adelson s recursion scheme for the computation of the compound Poisson probabilities rn (t), n 0; see Section 12 The best general-purpose approach for the computation of the state probabilities is the discrete FFT method An explicit expression for the generating function

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can be given It is a matter of tedious algebra to derive from (932) that P (z) = (1 ) where

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(933)

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The derivation uses that e {1 G(z)}t is the generating function of the compound Poisson probabilities rn (t); see Theorem 121 Moreover, the derivation uses that the generating function of the convolution of two discrete probability distributions is the product of the generating functions of the two probability distributions The other details of the derivation of (933) are left to the reader For constant and phase-type services, no numerical integration is required to evaluate the function (z) in the discrete FFT method Asymptotic expansion The state probabilities allow for an asymptotic expansion when it is assumed that the batch-size distribution and the service-time distribution are not heavy-tailed Let us make the following assumption

THE M X /G/1 QUEUE

Assumption 931 (a) The convergence radius R of G(z) = j =1 j zj is larger st than 1 Moreover, 0 e {1 B(t)} dt < for some s > 0 (b) lims B 0 e st {1 B(t)} dt = , where B is the supremum over all s with 0

e st {1 B(t)} dt] <

(c) limx R0 G(x) = 1 + B/ for some number R0 with 1 < R0 R Under this assumption we obtain from Theorem C1 in Appendix C that pj j as j , (934)

where is the unique solution to the equation ( ){1 G( )} = 1 on (1, R0 ) and the constant is given by = (1 )(1 ) ( ){1 G( )} A formula for the average queue size

The long-run average number of customers in queue is Lq = j =1 (j 1)pj Using the relation P (1) = j =1 jpj , we obtain after some algebra from (933) that

(935)

(1 )G ( ) +1 1 G( )

(936)

Lq =

E(X2 ) 1 2 2 (1 + cS ) + 1 , 2 1 2(1 ) E(X)

where X denotes the batch size Note that the rst part of the expression for Lq gives the average queue size in the standard M/G/1 queue, while the second part re ects the additional effect of the batch size The formula for Lq implies directly a formula for the long-run average delay in queue per customer By Little s formula Lq = Wq 932 The Waiting-Time Probabilities The concept of waiting-time distribution is more subtle for the case of batch arrivals than for the case of single arrivals Let us assume that customers from each arrival group are numbered as 1, 2, Service to customers from the same arrival group is given in the order in which those customers are numbered For customers from different batches the service is in order of arrival De ne the random variable Dn as the delay in queue of the customer who receives the nth service In the batch-arrival queue, limn P {Dn x} need not exist To see this, consider the particular case