In the present article, we analyze a class of time-limited polling systems. In particular, we will derive a direct relation for the evolution of the joint queue-length during the course of a server visit. This will be done both for the pure and the exhaustive exponential time-limited discipline for general service time requirements and preemptive service. More specifically, service of individual customers is according to the preemptive-repeat-random strategy, i.e., if a service is interrupted, then at the next server visit a new service time will be drawn from the original service-time distribution. Moreover, we incorporate customer routing in our analysis, such that it may be applied to a large variety of queueing networks with a single server operating under one of the before-mentioned time-limited service disciplines. We study the time-limited disciplines by performing a transient analysis for the queue length at the served queue. The analysis of the pure time-limited discipline builds on several known results for the transient analysis of the M/G/1 queue. Besides, for the analysis of the exhaustive discipline, we will derive several new results for the transient analysis of an M/G/1 during a busy period. The final expressions (both for the exhaustive and pure case) that we obtain for the key relations generalize previous results by incorporating customer routing or by relaxing the exponentiality assumption on the service times. Finally, based on the interpretation of these key relations, we formulate a conjecture for the key relation for any branching-type service discipline operating under an exponential time-limit.
|Name||Memorandum / Faculty of Mathematical Sciences|
|Publisher||University of Twente, Department of Applied Mathematics|