## 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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### References:

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