2계 마코프 도착과정의 분기 과정: 적률과 최소 표현 라플라스 변환에 관한 연구 - 김선교 (한국경영과학회지, 2024)
A split process from a Markovian arrival process of order n, MAP(n), is again a MAP(n). The transition rate matrices of the split process from a MAP is given in terms of the transition rate matrices of the original MAP. The moments of the split process can be determined if the transition rate matrices are given. In a typical queueing network analysis based on MAPs, the transition rate matrices of each arrival and departure process is approximated as a MAP based on moments. However, the conversion procedure from moments to a Markovian representation transition rate matrices involves iterative numerical procedures. In this paper, we study the minimal LT representation of the split process based on the one-to-one correspondence between n2 moments and the minimal Laplace transform (LT) coefficients of two consecutive stationary intervals. We show that the minimal LT coefficients of the split process from a MAP(2) can be represented in closed-form formula by those of the original process. Combined with the transformation between moments and the LT representation, our closed-form formula can be used to determine the moments of the split process from a MAP without the Markovian representation of the MAPs involved.
Keywords: Markovian Arrival Processes; Minimal Representation; Laplace Transform; Split Process