Introduction to Matrix-Analytic Methods in Queues 2

3Single-Server Queues Based on Arbitrary Epochs

In this chapter, we present ways to study queueing models by looking at arbitrary times. This approach is convenient and easy in some cases even though the dimensionality of the problem may increase. The type of queueing models to be studied via arbitrary time approach depends on whether one is dealing with M/G/1, GI/M/1 or quasi-birth-and-death (QBD) in continuous-time. It is worth pointing out that the QBD processes can be thought as the intersection of M/G/1-type and GI/M/1-type queues. Since this section relies heavily on Chapter 8 of Volume 1, readers who are not familiar with them are highly encouraged to review the materials from that chapter before pursuing further. The chapter is organized as follows. In section 3.1, we study BMAP/PH/1 queue, and in sections 3.2, 3.3, 3.4, 3.5 and 3.6, respectively, we look at the special cases of BMAP/PH/1, namely, MAP/PH/1, PH/PH/1, PH/M/1, M/PH/1 and M/M/1 queues.

3.1. BMAP/PH/1 queue

Consider a single-server queueing model in which the customers arrive according to a batch Markovian arrival process (BMAP) with representation given by {D_k} of order m. Let and denote δ to be the invariant vector of the irreducible generator = D₀ + D. That is, δ satisfies:

The batch arrival rate, λ