Introduction to Matrix Analytic Methods in Queues 1

5Discrete-Batch Markovian Arrival Process

In this chapter, the discrete-time batch Markovian arrival process (D-BMAP) is introduced. While developing the phase type (PH) distributions and the matrix-analytic methods (MAM) along with incorporating algorithmic methods in stochastic modeling, Neuts realized that the Markov renewal theory of M/G/1 type is applicable to many queueing and other models where the input processes retain certain Markovian features without being renewal processes (let alone Poisson processes). That led to the formulation of a versatile class of processes, initially referred to as the versatile Markovian point process (VMPP) (Neuts (1979)). Ramaswami in his doctoral dissertation, referring to VMPP as N-process, studied N/G/1 queue and its detailed analysis (Ramaswami 1980). The original description of the VMPP was quite involved with heavy notation due to the incorporation of various types of arrivals. Lucantoni et al. (1990) introduced the terms Markovian arrival process (MAP) and batch Markovian arrival process (BMAP) to describe the VMPP through simpler notation. Since then, these notations have become standard in describing the BMAP and the MAP. The reason for introducing the terms BMAP and MAP was at that time it was thought that the VMPP was a special class of the BMAP. However, it was realized later that the VMPP and the BMAP are equivalent processes (see Lucantoni (1991, 1993)). Thus, the BMAP (a.k.a. VMPP) is due to Neuts.

Earlier we saw how the ...