EURO Working Group on Stochastic Modelling

FIRST MEETING: 19-21 APRIL 2006, AMSTERDAM

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Abstracts

Rob van der Mei, Towards a unified theory on polling systems in heavy traffic

Polling systems are multi-queue systems in which a single server visits the queues in some order to serve customers waiting at the queues, typically incurring a switch-over time to move from one queue to the next. Polling models occur naturally in the modeling of applications in which different types of jobs compete for access to a common resource (e.g., CPU, bandwidth, labor). Typical application areas are computer-communication systems, flexible manufacturing systems and maintenance systems.
The key performance metrics are the waiting-time distributions at each of the queues. The proper operation of polling systems in terms of the waiting times incurred at each of the queues is particularly important when the system is heavily loaded. Motivated by this, in the literature for a variety of model instances heavy-traffic asymptotes of the waiting-time distributions have been derived. Despite the fact that these structure of distributions were found to be remarkably similar, so far a unifying theory that includes these model instances as special cases has been lacking.
In this presentation, I will present a unifying theory for the heavy-traffic limits of the waiting-time distributions for branching-type polling models. This theory leads to closed-form expressions for the LST of the waiting-time distributions in a very general parameter setting. The results are elegant and strikingly simple, and reveal explicitly how the asymptotic waiting-time distributions depend on the system parameters. The results not only include many known results as special cases, but also lead to new closed-form expressions for a variety of models that have been studied in the literature.