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The use of Spitzer's identity in the investigation of the busy period and other quantities in the queue GI/G/ 1

1962; Cambridge University Press; Volume: 2; Issue: 3 Linguagem: Inglês

10.1017/s1446788700026938

2059-9234

J. F. C. Kingmán,

Stochastic processes and statistical mechanics

As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queue M/G/ 1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queue GI/G/ 1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution in GI/G/ 1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the quene GI/E n / 1, as well as well-known results for M/G/ 1.