Stochastic Monotonicity of Markovian Multiclass Queueing Networks

Abstract

Multiclass queueing networks (McQNs) extend the classical concept of the Jackson network by allowing jobs of different classes to visit the same server. Although 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 multiclass 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 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 that are commonly encountered for "traditional" queueing processes, such as (i) monotonicity with respect to external arrival rates and (ii) starconvexity 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 simulationbased method that allows one to numerically determine the stability region of a McQN parameterized in terms of the arrival-rates vector.