Joint queue length distribution of multi-class, single server queues with preemptive priorities

In this paper we analyze an $M/M/1$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an $M/M/1$ queueing system with non-preemptive priorities between customer classes.Comment: 15 pages, 5 figures -- version 3 incorporates minor textual changes and fixes a few math typo

Joint queue length distribution of multi-class, single-server queues with preemptive priorities

In this paper we analyze an MN/MN/1 queueing system with N customer classes and preemptive priorities between classes, by using matrix-analytic techniques. This leads to an exact method for the computation of the steady state joint queue length distribution. We also indicate how the method can be extended to models with multiple servers and other priority rules

A polling model with an autonomous server

Polling models are used as an analytical performance tool in several application areas. In these models, the focus often is on controlling the operation of the server as to optimize some performance measure. For several applications, controlling the server is not an issue as the server moves independently in the system. We present the analysis for such a polling model with a so-called autonomous server. In this model, the server remains for an exogenous random time at a queue, which also implies that service is preemptive. Moreover, in contrast to most of the previous research on polling models, the server does not immediately switch to a next queue when the current queue becomes empty, but rather remains for an exponentially distributed time at a queue. The analysis is based on considering imbedded Markov chains at specific instants. A system of equations for the queue-length distributions at these instant is given and solved for. Besides, we study to which extent the queues in the polling model are independent and identify parameter settings for which this is indeed the case. These results may be used to approximate performance measures for complex multi-queue models by analyzing a simple single-queue model