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A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times. (English) Zbl 0566.60090
In this note we prove some stochastic decomposition results for variations of the GI/G/1 queue. Our main model is a GI/G/1 queue in which the server, when it becomes idle, goes on a vacation for a random length of time. On return from vacation, if it finds customers waiting, then it starts serving the first customer in the queue. Otherwise it takes another vacation and so on.
Under fairly general conditions the waiting time of an arbitrary customer, in steady state, is distributed as the sum of two independent random variables: one corresponding to the waiting time without vacations and the other to the stationary forward recurrence time of the vacation. This extends the decomposition result of E. Gelenbe and R. Iasnogorodski, Ann. Inst. Henri Poincaré, Nouv. Ser., Sect. B 16, 63- 73 (1980; Zbl 0433.60086). We use sample path arguments, which are also used to prove stochastic decomposition in a GI/G/1 queue with set-up time.

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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