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The existence of fixed points for the \(\cdot\)/GI/1 queue. (English) Zbl 1084.60057

Summary: A celebrated theorem of Burke’s asserts that the Poisson process is a fixed point for a stable exponential single server queue; that is, when the arrival process is Poisson, the equilibrium departure process is Poisson of the same rate. This paper considers the following question: Do fixed points exist for queues which dispense i.i.d. services of finite mean, but otherwise of arbitrary distribution (i.e., the so-called \(\cdot/\text{GI}/1/\infty/\text{FCFS}\) queues)? We show that if the service time \(S\) is nonconstant and satisfies \(\int\! P\{S \geq u\}^{1/2}\,du <\infty\), then there is an unbounded set \({\mathcal S}\subset(E[S],\infty)\) such that for each \(\alpha\in{\mathcal S}\) there exists a unique ergodic fixed point with mean inter-arrival time equal to \(\alpha\). We conjecture that in fact \({\mathcal S}=(E[S],\infty)\).

MSC:

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