ALGORITHMIC ANALYSIS OF QUEUEING MODELS in Visual Studio .NET

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ALGORITHMIC ANALYSIS OF QUEUEING MODELS
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Conditional waiting-time percentiles
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2 cS = 05 2 cS = 2
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02 0200 0167 0203 0282 0082 0067 0082 0146 0193 0167 0192 0218 0040 0033 0040 0048
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05 0569 0520 0580 0609 0240 0208 0243 0277 0554 0520 0556 0562 0118 0104 0119 0117
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099 332 345 331 333 137 138 136 136 342 345 342 342 0703 0691 0701 0690
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02 0256 0335 0264 0158 0099 0134 0104 0274 0335 0284 0232 0052 0067 0055 0038
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963 The M X /G/c Queue In the M X /G/c queue the customers arrive in batches rather than singly The arrival process of batches is a Poisson process with rate The batch size has a probability distribution { j , j = 1, 2, } with nite mean The service times of the customers are independent of each other and have a general distribution with mean E(S) There are c identical servers It is assumed that the server utilization , de ned by = E(S) , c
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is smaller than 1 The customers from different batches are served in order of arrival and customers from the same batch are served in the same order as their positions in the batch A computationally tractable analysis can only be given for the special cases of exponential services and deterministic services We rst analyse these two special cases Next we discuss a two-moment approximation for the general M X /G/c queue The M X /M/c queue The process {L(t)} describing the number of customers present is a continuoustime Markov chain Equating the rate at which the process leaves the set of states {i, i + 1, } to the rate at which the process enters this set of states, we nd for
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the state probabilities pj the recursion scheme
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min(i, c) pi =
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s ,
i = 1, 2, ,
(9632)
where = 1/E(S) Starting with p0 := 1, we successively compute p1 , p2 , and next obtain the desired pi by normalization The normalization can be based on Little s relation
c 1 c 1
jpj + c(1
j =0 j =0
pj ) = c
(9633)
stating that the average number of busy servers equals c The computational effort of the recursion scheme can be reduced by using the asymptotic expansion pj j as j , (9634)
where is the unique solution of the equation [1 ( )] = c (1 ) on the interval (1, R) and the constant is given by
(9635)
( 1) =
(c i)pi i /c (9636)
1 2 ( )/(c )
Here (z) = j =1 j zj and R is the convergence radius of the power series (z) To establish the asymptotic expansion, it is assumed that R > 1 In other words, the batch-size distribution is not heavy-tailed The derivation of the asymptotic expansion (9634) is routine De ne the generating function P (z) = j =0 pj zj , |z| 1 It is a matter of simple algebra to derive from (9632) that c 1
(1/c) P (z) =
(c i)pi zi
1 z{1 (z)}/{c (1 z)}
Next, by applying TheoremC1 in Appendix C, we obtain (9634) From the generating function we also derive after considerable algebra that the long-run average queue size is given by 1 Lq = c(1 )