# Adjoining batch Markov arrival processes of a Markov chain

Adjoining batch Markov arrival processes of a Markov chain A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Adjoining batch Markov arrival processes of a Markov chain

, Volume 34 (1) – Mar 8, 2018
10 pages

Publisher
Springer Journals
Copyright © 2018 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-018-0724-3
Publisher site
See Article on Publisher Site

### Abstract

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 8, 2018

### References

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