Large deviations ordering of point processes in some queueing networks

Abstract

Given a stochastic ordering between point processes, say that a p.p. N is smooth if it is less than the Poisson process with the same average intensity for this ordering. In this article we investigate whether initially smooth processes retain their smoothness as they cross a network of FIFO {\cdot}\,/D/1 queues along fixed routes. For the so-called strong variability ordering we show that point processes remain smooth as they proceed through a tandem of quasi-saturated (i.e., loaded to 1) M+{\cdot}\,/D/1 queues. We then introduce the Large Deviations ordering, which involves comparison of the rate functions associated with Large Deviations Principles satisfied by the point processes. For this ordering, we show that smoothness is retained when the processes cross a feed-forward network of unsaturated \cdot/D/1 queues. We also examine the LD characteristics of a deterministic p.p. at the output of an M+\cdot/D/1 queue.