A Numerical Approach to Stability of Multiclass Queueing Networks

The multiclass queueing network (McQN) arises as a natural multiclass extension of the traditional (single-class) Jackson network. In a single-class network, subcriticality (i.e., subunitary nominal workload at every station) entails stability, but this is no longer sufficient when jobs/customers of different classes (i.e., with different service requirements and/or routing scheme) visit the same server; therefore, analytical conditions for stability of McQNs are lacking, in general. In this note, we design a numerical (simulation-based) method for determining the stability region of a McQN, in terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity properties enjoyed by the associated Markovian queue-configuration process. Stochastic monotonicity is a quite common feature of queueing models and can be easily established in the single-class framework (Jackson networks); recently, also for a wide class of McQNs, including first-come-first-serve networks, monotonicity properties have been established. Here, we provide a minimal set of conditions, under which the method performs correctly. Eventually, we illustrate the use of our numerical method by presenting a set of numerical experiments, covering both single- and multiclass networks.