Stochastic Comparison of Queueing Networks

Abstract

We recall classical queueing networks and their stochastic monotonicity properties as a special case of a general stochastic ordering theory for Markov processes. As a consequence of stochastic monotonicity we present stochastic bounds in transient and stationary conditions for the queue length processes, and some dependence and ordering properties for sojourn times in networks.We overview properties of throughputs in networks in connection with stochastic orderings. Finally we concentrate on dependence orderings for queueing networks with a special attention on the role of routing as a parameter influencing correlation structures in networks. Some connections to the problem of speed of convergence to stationarity via spectral gaps are pointed out.