An exact solution for the state probabilities of the multi-class, multi-server queue with preemptive priorities

We consider a multi-class, multi-server queueing system with preemptive priorities. We distinguish two groups of priority classes that consist of multiple customer types, each having their own arrival and service rate. We assume Poisson arrival processes and exponentially distributed service times. We derive an exact method to estimate the steady state probabilities. Because we need iterations to calculate the steady state probabilities, the only error arises from choosing a finite number of matrix iterations. Based on these probabilities, we can derive approximations for a wide range of relevant performance characteristics, such as the moments of the number of customers of a certain type in the system en the expected postponement time for each customer class. We illustrate our method with some numerical examples. Numerical results show that in most cases we need only a moderate number of matrix iterations (~20) to obtain an error less than 1% when estimating key performance characteristics.