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 $\fF=(E[S],\infty)$.
Publié le : 2003-10-14
Classification:
Queue,
tandem queueing networks,
general independent services,
stability,
Loynes theorem,
Burke theorem,
60K25,
60K35,
68M20,
90B15,
90B22
@article{1068646383,
author = {Mairesse, Jean and Prabhakar, Balaji},
title = {The existence of fixed points for the $\cdot/GI/1$ queue},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 2216-2236},
language = {en},
url = {http://dml.mathdoc.fr/item/1068646383}
}
Mairesse, Jean; Prabhakar, Balaji. The existence of fixed points for the $\cdot/GI/1$ queue. Ann. Probab., Tome 31 (2003) no. 1, pp. 2216-2236. http://gdmltest.u-ga.fr/item/1068646383/