Markovian arrivals in stochastic modelling: a survey and some new results

This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension.Peer Reviewe

AN EXTENSION OF THE MATRIX-ANALYTIC METHOD FOR M/G/1-TYPE MARKOV PROCESSES

Abstract We consider a bivariate Markov process {(U (t), S(t)); t ≥ 0}, where U (t) (t ≥ 0) takes values in [0, ∞) and S(t) (t ≥ 0) takes values in a finite set. We assume that U (t) (t ≥ 0) is skip-free to the left, and therefore we call it the M/G/1-type Markov process. The M/G/1-type Markov process was first introduced as a generalization of the workload process in the MAP/G/1 queue and its stationary distribution was analyzed under a strong assumption that the conditional infinitesimal generator of the underlying Markov chain S(t) given U (t) > 0 is irreducible. In this paper, we extend known results for the stationary distribution to the case that the conditional infinitesimal generator of the underlying Markov chain given U (t) > 0 is reducible. With this extension, those results become applicable to the analysis of a certain class of queueing models. Keywords: Queue, bivariate Markov process, skip-free to the left, matrix-analytic method, reducible infinitesimal generator, MAP/G/1 queue Introduction We consider a bivariate Markov process {(U (t), S(t)); t ≥ 0}, where U (t) and S(t) are referred to as the level and the phase, respectively, at time t. U (t) (t ≥ 0) takes values in [0, ∞) and S(t) (t ≥ 0) takes values in a finite set M = {1, 2, . . . , M }. {U (t); t ≥ 0} either decreases at rate one or has upward jump discontinuities, so that {U (t); t ≥ 0} is skip-free to the left. We assume that when (U (t−), S(t−)) = (x, i) (x > 0, i ∈ M), an upward jump (possibly with size zero) occurs at a rate σ [i] (σ [i] > 0) and the phase S(t) becomes j (j ∈ M) with probability p [i,j] . On the other hand, when (U (t−), S(t−)) = (0, i) (i ∈ M), an upward jump occurs with probability one and the phase S(t) becomes j (j ∈ M) with probability p [i,j] . Note here that for i ∈ M,

Markovian arrivals in stochastic modelling : a survey and some new results

This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension