Explicit and Iterative Numerical Approaches to Solving Queueing Models

This paper considers two approaches to the numerical solution of single node queueing models. Both approaches use a phase-type distribution to model very general service processes. The first approach is explicit and sometimes can exploit the structure of certain balance equations to reduce the global balance equations from a set of second order difference equations to a set of first order difference equations. This reduction permits the steady-state probability distribution to be written explicitly as a function of the model parameters. The second approach, due to Neuts, uses the fact that most queues that have a matrix-geometric steady-state probability distribution can be solved by means of a recursive technique. We compare the approaches and present a theorem that specifies necessary conditions for the computation of an explicit solution. A number of examples are provided.