6Continuous-Batch Markovian Arrival Process
In this chapter, our focus is on the continuous-time version of D-BMAP. We refer to this as a continuous-time batch Markovian arrival process, abbreviated as BMAP. Unlike the notation for the discrete case, we prefer not to use C-BMAP mainly because the rest of the two-volume book will focus on continuous-time models (except when dealing with embedded processes) and it is consistent with the published articles in the literature to call it BMAP when dealing with continuous-time BMAP. This chapter is organized as follows. In section 6.1, we present the basics of BMAP, and in section 6.2 we discuss the counting process associated with BMAP. In section 6.3, we present a way to generate BMAP processes for numerical purposes, especially when one wants to study queueing models to be discussed in Volume 2.
6.1. Continuous-time batch Markovian arrival process (BMAP)
Suppose we consider an irreducible continuous-time Markov chain (CTMC) with m transient states and one absorbing state. Suppose that the initial probability vector of this CTMC is given by ξ and Q = (qij) is the generator. At the end of a sojourn time in state i, that is, exponentially distributed with parameter λi ≥ −qi, i, one of the following two events could occur: with probability pij (k) the transition corresponds to an arrival of group size k, k ≥ 1, and the underlying Markov chain is in state j with 1 ≤ i, j ≤ m; with probability pij(0) the transition corresponds to no ...