2 2 Papp = (1 cS )PGI /D/c + cS PGI /M/c

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(975)

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2 provided cS is not too large and the traf c load on the system is not very light In this formula PGI /D/c and PGI /M/c denote the exact values of the speci c performance measure for the special cases of the GI /D/c queue and the GI /M/c queue with the same mean service time E(S) Table 971 gives for several values

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ALGORITHMIC ANALYSIS OF QUEUEING MODELS

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Table 971 Lq exa app exa app 0066 0082 0006 0009

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Some numerical results for the E10 /E2 /c queue = 05 = 08 = 09 Lq 0780 0813 0452 0466 (08) (095) 259 257 0551 0530 478 476 0993 0968 Lq 221 225 175 176 (08) (095) 499 514 102 102 925 925 187 186

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c 1 5

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(08) (095) 121 119 0277 0243 221 217 0499 0452

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of c and the exact and approximate values of the average queue size Lq and the conditional waiting-time percentiles (08) and (095) for the E10 /E2 /c queue In all examples the normalization E(S) = 1 is used The above linear interpolation formula is in general not to be recommended for the delay probability, particularly 2 not when cS is close to zero For example, the delay probability has the respective values 00776, 03285 and 03896 for the E10 /D/5 queue, the E10 /E2 /5 queue and the E10 /M/5 queue, each with = 08 Interpolation formulas like the one above should always be accompanied by a caveat against their blind application The above interpolation formula re ects the empirical nding that measures of system performance are in general much more sensitive to the interarrival-time distribution than to the service-time distribution, in particular when the traf c load is light 971 The GI /M/c Queue In the GI /M/c queue the service times of the customers are exponentially distributed with mean 1/ In addition to the time-average probabilities pj , let j = the long-run fraction of customers who nd j other customers present upon arrival There is a simple relation between the pj and the j We have min(j, c) pj = j 1 , j = 1, 2, (976)

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This relation equates the average number of downcrossings from state j to state j 1 per time unit to the average number of upcrossings from state j 1 to state j per time unit; see also Section 27 The probabilities j determine the waiting-time distribution function Wq (x) Note that the conditional waiting-time of a customer nding j c other customers present upon arrival is the sum of j c + 1 independent exponentials with mean 1/(c ) and thus has an Erlang distribution Hence, by conditioning,

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j c

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1 Wq (x) =

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j =c

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e c x

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(c x)k , k!

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x 0

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(977)

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THE GI /G/c QUEUE

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This expression can be further simpli ed To show this, we use that j +1 = , j j c 1 (978)

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for some constant 0 < < 1 The proof of this result is a replica of the proof of the corresponding result for the GI /M/1 queue; see (3515) Hence j = j c+1 c 1 , j c 1 (979)

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As a by-product of (976) and (977) we have pj = j c pc , Substituting (979) into (978) yields 1 Wq (x) = c 1 e c (1 )x , 1 x 0 (9711) j c (9710)

The constant is the unique solution of the equation =

e c (1 )t a(t) dt

(9712)

on the interval (0,1) To see this, note that { j } is the equilibrium distribution of the embedded Markov chain describing the number of customers present just before an arrival epoch Substituting (979) into the balance equations

j =

k=j 1

e c t

(c t)k+1 j a(t) dt, (k + 1 j )!

j c

easily yields the result (9712) By the relations (976), (979) and (9710), the probability distributions {pj } and { j } are completely determined once we have computed 0 , , c 1 or p0 , , pc These c unknowns can be rather easily computed for the special cases of deterministic, Coxian-2 and Erlangian interarrival times If one is only interested in the waiting-time probabilities (9711), these computations can be avoided a An explicit expression for the delay probability c 1 /(1 ) is given in Tak cs (1962) For the case of c = 1 (GI /M/1 queue), c 1 /(1 ) = Deterministic arrivals Suppose there is a constant time D between two consecutive arrivals De ne the embedded Markov chain {Xn } by Xn = the number of customers present just before the nth arrival