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October 2003 The existence of fixed points for the $\cdot/GI/1$ queue
Jean Mairesse, Balaji Prabhakar
Ann. Probab. 31(4): 2216-2236 (October 2003). DOI: 10.1214/aop/1068646383


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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Jean Mairesse. Balaji Prabhakar. "The existence of fixed points for the $\cdot/GI/1$ queue." Ann. Probab. 31 (4) 2216 - 2236, October 2003.


Published: October 2003
First available in Project Euclid: 12 November 2003

zbMATH: 1084.60057
MathSciNet: MR2016617
Digital Object Identifier: 10.1214/aop/1068646383

Primary: 60K25, 60K35, 68M20, 90B15, 90B22

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 4 • October 2003
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