Generalized phase-type distributions based on multi-state systems

It is a well-known definition that the time until entering an absorbing state in a finite state Markov process follows a phase-type distribution. In this article we extend this distribution through adding two new events: one is that the number of transitions among states reaches a specified threshold; the other is that the sojourn time in a specified subset of states exceeds a given threshold. The system fails when it enters the absorbing state or two new events happen, whichever occurs first. We develop three models in terms of three circumstances: (i) the two thresholds are constants; (ii) the number of transitions is random while the sojourn time in the specified states is constant; and (iii) the sojourn time in the specified states is random while the number of transitions is constant. To the performance of such systems, we employ the theory of aggregated stochastic processes and obtain closed-form expressions for all reliability indexes, such as point-wise availabilities, various interval availabilities, and distributions of lifetimes. We select special distributions of two thresholds for models 2 and 3, which are the exponential and geometric distributions. The corresponding formulas are presented. Finally, some numerical examples are given to demonstrate the proposed formulas.