Stochastic Monotonicity of Markovian Multi-class Queueing Networks

Abstract

Multi-class queueing networks (McQNs) extend the classical concept of Jackson network by allowing jobs of different classes to visit the same server. While such a generalization seems rather natural, from a structural perspective there is a significant gap between the two concepts. Nice analytical features of Jackson networks, such as stability conditions, product-form equilibrium distributions, and stochastic monotonicity do not immediately carry over to the multi-class framework. The aim of this paper is to shed some light on this structural gap, focusing on monotonicity properties. To this end, we introduce and study a class of Markov processes, which we call \emph{Q-processes}, modeling the time evolution of the network configuration of any open, work-conservative McQN having exponential service times and {Poisson input}. We define a new monotonicity notion tailored for this class of processes. Our main result is that we show monotonicity for a large class of McQN models, covering virtually all instances of practical interest. This leads to interesting properties which are commonly encountered for `traditional' queueing processes, such as (i)~monotonicity with respect to external arrival rates and (ii)~star-convexity of the stability region (with respect to the external arrival rates); such properties are well known for Jackson networks, but had not been established at this level of generality. This research was partly motivated by the recent development of a simulation-based method which allows one to numerically determine the stability region of a McQN parametrized in terms of the arrival rates vector.