A Finite Difference Equations Approach to a Priority Queue

The main goal of this note is to demonstrate the use of a direct approach, as opposed to generating functions, for solving balance equations of a queueing system whose state is described by two independent variables. The basic idea is to consider a function of two variables as a set of functions of one independent variable. Hence the balance equations may be viewed as a set of simultaneous difference equations, and solved using the appropriate techniques. This method may allow the inclusion of queue-dependent service and arrival rates, and an easy treatment of finite queueing rooms. This is illustrated in the example of a two-level preemptive priority queue with exponentially distributed interarrival and service times.