Asymptotic behaviors of aggregated Markov processes

Finite state Markov processes and their aggregated Markov processes have been extensively studied, especially in ion channel modeling and reliability modeling. In reliability field, the asymptotic behaviors of repairable systems modeled by both processes have been paid much attention to. For a Markov process, it is well-known that limiting measures such as availability and transition probability do not depend on the initial state of the process. However, for an aggregated Markov process, it is difficult to directly know whether this conclusion holds true or not from the limiting measure formulas expressed by the Laplace transforms. In this paper, four limiting measures expressed by Laplace transforms are proved to be independent of the initial state through Tauber’s theorem. The proof is presented under the assumption that the rank of transition rate matrix is one less than the dimension of state space for the Markov process, which includes the case that all states communicate with each other. Some numerical examples and discussions based on these are presented to illustrate the results directly and to show future related research topics. Finally, the conclusion of the paper is given.