# Modeling a Multiprocessor System with Preemptive Priorities

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## Abstract

Consider a system with N processors and two job types with one having preemptive priority over the other. Arrivals are Poisson and service times are exponential. We present two approaches for modeling the system. The difference between the two lies in the order with which we list the numbers of jobs of each type in the system in the state definition. Although both approaches yield matrix-geometric solutions, their implications for computation are significantly different. The first approach has attractive structural properties and an easily solvable rate matrix, but the system of equations at the boundary is most often prohibitively large. The second approach, while not amenable to any easy computation of its rate matrix, renders itself to an efficient solution at the boundary, and provides a basis for waiting time analysis for low priority jobs. In the paper, we present an efficient implementation of state reduction for solving the stationary probabilities associated with the boundary states. We give a numerical example to highlight various issues in the computer solution.

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