- Author:
- Sankaran Kumar Narayanan,
- Chair:
- Edward P. Kao
In this research, we consider the following M/M/N system with servers' vacations. A server that completes service leaves the system for a vacation whose duration follows an exponential distribution. We examine two models--multiple vacations model and single vacations model. In the first, a server returning to an empty queue takes another vacation immediately. In the second, the returning server stays in the system to serve at least one customer before leaving on another vacation.
We use the matrix-geometric approach to model the multiple vacations model and propose algorithms for computing the stationary queue length distribution, the expected length of busy period, the mean waiting time, and the waiting time distribution. We illustrate our algorithms with numerical examples at the end. We show that the single vacations model can be treated as a variation of the multiple vacations case and derive the results for it from the procedures developed for solving the earlier model.
In contrast to the vacation models with a single server, relatively little work has been done on multi-server queues. The two models mentioned here were studied by Levi and Yechiali (1974). They presented a transform approach for finding the distribution of busy servers and the mean queue length. Mitrani and Avi-Itzhak (1968) examined a two-server service breakdown model. This was extended to the multi-server case by Neuts and Lucantoni (1979) using matrix-geometric approach.
We also investigate a multi-server priority queueing system with both preemptive and non-preemptive priorities and heterogeneous service rates. Arrivals occur at a Poisson rate and service time distributions are assumed to be exponential. We propose a solution procedure using the matrix-geometric approach. We develop numerical procedures for computing the steady-state probabilities and other queue characteristics. The preemptive priority model was first analyzed by Mitrani and King (1981) using geometric transforms. Miller (1981, 1982) used the matrix-geometric approach to study the M/M/1 and M/M/N priority queue. Recently Gail, Hantler, and Taylor (1988) examined the non-preemptive priority queue to obtain generating functions for the equilibrium probability distribution. (Abstract shortened with permission of author.)
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