1 College of Mathematics and Statistics, Hunan University of Finance and Economics, Changsha 410205, China; 2 College of Mathematics and Computational Science, Hunan University of Arts and Science, Changde 415000, China; 3 Key Laboratory of High performance Computing and Stochastic Information Processing, Ministry of Education of China, College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China; 4 Hunan Province Cooperative Innovation Center for the Construction and Development of Dongting Lake Ecological Economic Zone, Changde 415000, China

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.

Fund:Supported by the National Natural Science Foundation of China (No.11671132, 11601147), Hunan Provincial Natural Science Foundation of China (No.16J3010), Philosophy and Social Science Foundation of Hunan Province (No.16YBA053) and Key Scientific Research Project of Hunan Provincial Education Department (No.15A032).

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