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 i...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.展开更多
基金Supported by the National Natural Science Foundation of China(No.11671132,11601147)Hunan Provincial Natural Science Foundation of China(No.16J3010)+1 种基金Philosophy and Social Science Foundation of Hunan Province(No.16YBA053)Key Scientific Research Project of Hunan Provincial Education Department(No.15A032)
文摘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.
