Phase‐type models involving restarting and instantaneous transitions, with applications to degradation and maintenance

Abstract

A phase‐type distribution is the distribution of time to absorption for an absorbing continuous‐time finite state Markov chain. The paper first reviews the extension of the phase‐type setting to modeling of competing risks by introducing multiple absorbing states. The main study of the paper is the further extension to introducing instantaneous transitions at certain stages of the original models. The motivation is from applications to repair and maintenance, bringing failed systems into working ones by instantaneous repair actions. Two slightly different approaches are studied. The first one is based on restarting the original Markov chain upon absorption, leading to the consideration of a Markov renewal process. The second approach involves periodically inspected systems, where maintenance actions are modeled by instantaneous transitions made at regular inspection times. For both approaches are suggested measures of reliability and maintenance based on long run properties.

Preface to the special issue on degradation and maintenance, modelling and analysis

“…This fact makes this distribution very attractive as parametric model to failure time data with degradation trend. In this sense, Lindqvist 6 shows how phase-type distributions can be used for modelling the effect of degradation and maintenance. The paper focuses on Phase-type models used in competing risks framework who consider multiple absorbing states on their continuous time Markov chain.…”