Efficient Algorithmic Solutions to Exponential Tandem Queues with Blocking

Abstract

Stable queuing systems consisting of two groups of servers, having exponential service times, placed in tandem and separated by a finite buffer, are shown to have a steady-state probability vector of matrix-geometric form. The queue is stable as long as the Poisson arrival rate does not exceed a critical value, which depends in a complicated manner on the service rates, the numbers of servers in each group, the size of the intermediate buffer and the unblocking rule followed when system becomes blocked. The critical input rate is determined in a unified manner. For stable queues, it is shown how the stationary probability vector and other important features of the queue may be computed. The essential step in the algorithm is the evaluation of the unique positive solution of a quadratic matrix equation.