Abstract
The class of counting processes constitutes a significant part of applied probability. The classic counting processes include Poisson processes, nonhomogeneous Poisson processes, and renewal processes. More sophisticated counting processes, including Markov renewal processes, Markov modulated Poisson processes, age-dependent counting processes, and the like, have been developed for accommodating a wider range of applications. These counting processes seem to be quite different on the surface, forcing one to understand each of them separately. The purpose of this paper is to develop a unified multivariate counting process, enabling one to express all of the above examples using its components, and to introduce new counting processes. The dynamic behavior of the unified multivariate counting process is analyzed, and its asymptotic behavior ast→∞is established. As an application, a manufacturing system with certain maintenance policies is considered, where the optimal maintenance policy for minimizing the total cost is obtained numerically.
Highlights
A stochastic process {N t : t ≥ 0} is called a counting process when N t is nonnegative, right continuous and monotone nondecreasing with N 0 0
Both Markov-Modulated Poisson Process (MMPP) and SMMPP will be proven to be expressible in terms of the components of the unified multivariate counting process proposed in this paper
A unified multivariate counting process M t, N t is proposed with nonhomogeneous Poisson processes lying on a finite semi-Markov process
Summary
A stochastic process {N t : t ≥ 0} is called a counting process when N t is nonnegative, right continuous and monotone nondecreasing with N 0 0. In the SMMPP, the counting process under consideration is modulated according to state transitions of the underlying semi-Markov process. The purpose of this paper is to develop a unified multivariate counting process which would contain all of the above examples as special cases In this regard, we consider a system where items arrive according to an NHPP. A manufacturing system with certain maintenance policies is considered, where the unified multivariate counting process enables one to determine numerically the optimal maintenance policy for minimizing the total cost. Detailed description of the unified multivariate counting process is provided in Section 3 and its dynamic behavior is analyzed in Section 4 by examining the probabilistic flow of the underlying stochastic processes and deriving transform results involving Laplace transform generating functions. A submatrix of a is defined by a GB aij i∈G,j∈B
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