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# Some Comparability Results for Waiting Times in Single- and Many-Server Queues

D. J. Daley and T. Rolski
Journal of Applied Probability
Vol. 21, No. 4 (Dec., 1984), pp. 887-900
DOI: 10.2307/3213704
Stable URL: http://www.jstor.org/stable/3213704
Page Count: 14
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## Abstract

It is shown that the stationary waiting time random variables W′, W′′ of two M/G/1 queueing systems for which the corresponding service time random variables satisfy $E\, (S^\prime-x)_+ \leqq E\, (S^{\prime\prime} - x)_+\, ({all}\, \, x > 0)$, are stochastically ordered as W′ ≤d W′′. The weaker conclusion, that E (W′-x)+ ≤ E (W′′-x)+, (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E (x-T′)+ ≤ E (x-T′′) + (all x).A sufficient condition for $w_k \equiv EW$ in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.

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