Some asymptotic properties of two-stage tandem networks of PH/PH/1 queues

Abstract

For a two-stage tandem network of PHPH queues, Fujimoto and Takahashi observed that the steady-state joint queue-length distribution has an asymptotic product-form as the number of customers in the system becomes large. Moreover, they noted that when the traffic intensity at the second queue is large, there appear to be two different asymptotic regimes. In this paper we give a detailed examination of the asymptotic properties of such a tandem queue by constructing two quasi-birth-and-death QBD models and provide an explanation for the results of Fujimoto and Takahashi. The matrix-analytic approach provides a unified framework which explains the asymptotic behaviour of the steady-state joint queue-length probabilities. The approach in this paper can be extended to general doubly-infinite QBDs and to multi-dimensional QBDs.