Stability of Join-the-Shortest-Queue Networks
J. G. Dai, J. J. Hasenbein, Bara Kim
This paper investigates stability behavior in a variant of a generalized
Jackson
queueing network. In our network, some customers use a
join-the-shortest-queue
policy when entering the network or moving to the next station.
Furthermore,
we allow interarrival and service times to have general distributions.
For
networks with two stations we derive necessary and sufficient conditions
for
positive Harris recurrence of the network process. These conditions
involve only
the mean values of the network primitives. We also provide
counterexamples showing
that more information on distributions and tie-breaking probabilities is
needed
for networks with more than two stations, in order to characterize the
stability
of such systems. However, if the routing probabilities in the network
satisfy a
certain homogeneity condition, then we show that the stability behavior
can
be explicitly determined, again using the mean value parameters of the
network.
A byproduct of our analysis is a new method for using the fluid model of
a queueing network to show non-positive recurrence of a
process.
In previous work, the fluid model was only used to show either positive
Harris recurrence or transience of a network process.