Introduction to Matrix-Analytic Methods in Queues 2

5Multi-Server Queues

Multi-server queueing systems are very common and useful in practice. While BMAP/PH/c and GI/PH/c queues can be studied using the techniques seen in the first three chapters of this volume, the restriction arises through the explosion of the state space. One needs to develop efficient algorithmic procedures to obtain the system performance measures. In this chapter, we discuss a few multi-server queueing models using the tools seen in those chapters along with algorithmic procedures by exploiting the coefficient matrices, and a few complex models will be analyzed through simulation in Chapter 7.

This chapter is organized as follows. In section 5.1, we discuss the complexity involved in the study of BMAP/PH/c queues. In the context of MAP/PH/3 queues with heterogeneous servers, we present in section 5.2 the steady-state analysis in great detail along with a number of illustrative examples. The discussion of BMAP/M/c queues is presented in section 5.3, and in section 5.4 a brief remark is made on GI/PH/c queues as detailed illustrative examples on this along with others will be presented in the chapter dealing with simulation.

5.1. BMAP/PH/c queue

Suppose we look at a c-server system in which arrivals occur according to a batch Markovian arrival process (BMAP) with representation matrices {D_k} of dimension m. First, we look at the heterogeneous server case and thus the service times are of continuous phase type (CPH) with representation (β(r), S(r)) of dimension ...