A Note on Computing Approach Toward Two-tier Service Models

Abstract

This paper presents a new algorithm for computing the performance measure of a two-tier service queueing model. In such a system, one service provider offers service with unlimited waiting space and the other offers a finite waiting space. Due to the two queue feature, the system is formulated as a state dependent quasi-birth-and-death (QBD) process. The customer choice behavior and the observable queues make the QBD process to have a large number of boundary states. Such a structure motivates us to develop a more efficient algorithm than the classical rate matrix iteration algorithms. With the special structure of the infinitesimal generator matrix for the two-tier service system, we propose a more efficient and innovative K-matrix based algorithm for computing the stationary distribution. As the buffer size increases, the improved accuracy and computational efficiency of the K-matrix method become  significant compared with the classical Geometric-Matrix method. We demonstrate the advantages of the new algorithm with numerical examples.