Stability Criteria for Multi-class Queueing Networks with Re-entrant Lines

In this paper we show how the stability criteria for the model proposed in MacPhee and Müller (Queueing Syst 52(3):215–229, 2006) can be applied to queueing networks with re-entrant lines. The model considered has Poisson arrival streams, servers that can be configured in various ways, exponential service times and Markov feedback of completed jobs. The stability criteria are expressed in terms of the mean drifts of the process under the various server configurations. For models with re-entrant lines we impose here a boundary sojourn condition to ensure adequate control of the process when one or more queues are empty. We show with some examples, including the generalised Lu–Kumar network discussed in Niño-Mora and Glazebrook (J Appl Probab 37(3):890–899, 2000), how our results can be applied.