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Table 981
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Numerical results for Prej in the M/G/c/c + N queue (c = 5) = 05 = 08 N =1 N =5
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where = E(S)/c and the constants an and bn are the same as in Theorem 961 Proof The proof of the theorem is a minor modi cation of the proof of Theorem 961 The details are left to the reader The result of Theorem 981 is exact for both the case of multiple servers with exponential service times and the case of a single server with general service times, since for these two special cases the approximation assumption holds exactly Further support for the approximate result of the theorem is provided by the fact that the approximation is exact for the case of no waiting room (N = 0) Numerical investigations indicate that the approximation for the state probabilities is accurate enough for practical purposes Table 981 gives the exact and approximate values of the rejection probability Prej for several examples The probability Prej denotes the long-run fraction of customers who are rejected By the PASTA property, Prej = pN +c
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2 In all examples we take c = 5 servers Deterministic services (cS = 0), E2 services 2 = 1 ) and H services with gamma normalization (c2 = 2) are considered For (cS 2 S 2 the latter two services, the exact values of Prej are taken from the tabulations of Seelen et al (1985) For deterministic services, computer simulation was used to nd Prej In the table we give the 95% con dence intervals It is interesting to point out that the results in Table 981 support the long-standing conjecture for the GI /G/c/c + N queue that Prej 1 1/ as N when > 1
A proportionality relation For the case of < 1 the computational work can be considerably reduced when the approximation to Prej must be computed for several values of N Denote by
ALGORITHMIC ANALYSIS OF QUEUEING MODELS
( ) ( ) pj (app) the approximation given in Theorem 961 to the state probability pj in the in nite-capacity M/G/c queue This approximation requires that < 1 An inspection of the recursion schemes in Theorems 961 and 981 reveals that, for some constant ,
( ) = pj (app),
j = 0, 1, , N + c 1
(981)
( ) The constant is given by = [1 j =N +c pj (app)] 1 In the next section it will be seen that this proportionality relation implies ( ) pj (app)
(1 ) Prej = 1
j =N +c app app app j =N +c
(982)
( ) pj (app)
( ) probabilities pj (app) was discussed in Section 962 app
where Prej = pN +c denotes the approximation to Prej The computation of the
( ) The approximations pj and pj (app) are exact both for the case of multiple servers with exponential service times and for the case of a single server with general service times Therefore relations (981) and (982) hold exactly for the M/M/c/c + N queue and the M/G/1/N + 1 queue For these particular queueing models the proportionality relation (981) can be directly explained by a simple probabilistic argument This will be done in the next subsection It is noted that for the general M/G/c/c + N queue the proportionality relation is not satis ed when ( ) the exact values of pj and pj are taken instead of the approximate values
982 A Basic Relation for the Rejection Probability In this section a structural form will be revealed for the rejection probability In many situations the rejection probability can be expressed in terms of the state ( ) denote probabilities in the in nite-capacity model In the following, pj and pj the time-average state probabilities for the nite-capacity model and the in nite( ) capacity model To ensure the existence of the probabilities pj , it is assumed that the server utilization is smaller than 1 Theorem 982 Both for the M/M/c/c + N queue and the M/G/1/N + 1 queue it holds that ( ) (983) pj = pj , j = 0, 1, , N + c 1 for some constant > 0 The constant is given by = [1 and the rejection probability is given by
j =N +c ( ) pj ] 1