Analysing sojourn times in queueing networks: A structural approach

Structural properties of generalised semi-Markov processes (GSMP) have been successfully studied in the literature. Examples are the celebrated commuting condition (CC), which is the key condition for unbiasedness of the infinitesimal perturbation analysis (IPA) gradient estimator, or the monotonicity condition (M), which implies stochastic order properties. When coming to queueing networks, these properties can be deduced from simple structural conditions on the topology of the network. Generally speaking, the results known so far apply to event times in GSMPs. Unfortunately, for multi or infinite server queueing networks, event times cannot be translated into customer related performance measures, such as sojourn times or waiting times. To overcome this drawback, we introduce in this paper a new technique, called "compound events". Compound events enable us to define event times in such a way that results with respect to event times can be translated into results with respect to customer related performance measures. We model a generic queueing network by means of a GSMP with compound events and we establish conditions on the topology of the queueing network that imply that the corresponding GSMP satisfies structural conditions like (CC) and (M). These structural conditions then imply properties of event times, like continuity or stochastic monotonicity. Using compound events, these results can be translated into properties of customer times, like sojourn times or waiting times. Our results extend the area of applicability of IPA to the analysis of customer related performance measures in multi-server queueing networks. Furthermore, we obtain new results on stochastic ordering of customer related performance measures.