Annals of Probability

The existence of fixed points for the $\cdot/GI/1$ queue

Jean Mairesse and Balaji Prabhakar

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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 $\cdot/\mathit{GI}/1/\infty/\mathit{FCFS}$ queues)? We show that if the service time $S$ is nonconstant and satisfies \mbox{$\int\!\! P\{S \geq u\}^{1/2}\,du <\infty$}, then there is an unbounded set $\fF\subset(E[S],\infty)$ such that for each $\alpha\in\fF$ there exists a unique ergodic fixed point with mean inter-arrival time equal to $\alpha$. We conjecture that in fact $\fF=(E[S],\infty)$.

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Ann. Probab., Volume 31, Number 4 (2003), 2216-2236.

First available in Project Euclid: 12 November 2003

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Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20]

Queue tandem queueing networks general independent services stability Loynes theorem Burke theorem


Mairesse, Jean; Prabhakar, Balaji. The existence of fixed points for the $\cdot/GI/1$ queue. Ann. Probab. 31 (2003), no. 4, 2216--2236. doi:10.1214/aop/1068646383.

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