Repairable models with operating and repair times governed by phase type distributions

Abstract

We consider a device that is subject to three types of failures: repairable, non-repairable and failures due to wear-out. This last type is also non-repairable. The times when the system is operative or being repaired follow phase type distributions. When a repairable failure occurs, the operating time of the device decreases, in that the lifetimes between failures are stochastically decreasing according to a geometric process. Following a non-repairable failure or after a previously fixed number of repairs occurs, the device is replaced by a new one. Under these conditions, the functioning of the device can be modelled by a Markov process. We consider two different models depending on whether or not the phase of the operational system at the instants of failure is remembered or not. For both models we derive the stationary distribution of the Markov process, the availability of the device, the rate of occurrence of the different types of failures, and certain quantities of interest.

A reliability semi‐Markov model involving geometric processes

“…Such is the case for the operational time following phase-type distribution. It can be shown that, in this case, the resulting expressions are the ones calculated in the paper of Neuts et al [3]. Moreover, the class of phase-type distributions have as subclasses some of frequent use in reliability: exponential, Erlangian, generalized Erlangian, and the hyperexponential.…”

“…Lam [2] calculated the rate of occurrence of failures for Markov systems and applied it to repairable systems. Neuts et al [3] considered a repairable system that included different types of failure, phase-type distributed operational and repair times, and a policy of repair N ; (the system is replaced at the…”

Performance evaluation of transfer lines with general repair times and multiple failure modes

“…For a review of methods using PH distributions in survival models see (Aalen 1995). In (Neuts et al 2000), a single machine is considered with multiple up and down states, in which both life time and repair time are PH distributed, showing that this distribution is particularly suitable for analyzing the maintenance of degrading processes and the reliability of reparable systems. The study in (Jafari and Shantikumar 1987) proposes both an exact and an approximate solution for a two-stage system with machines having generally distributed up and down times.…”

Shock and wear models under policy N using phase-type distributions

“…This policy has been applied in Ref. [2]. This procedure is a warranty for eluding dangerous damage.…”

“…The interarrival time between consecutive shocks follow continuous phase-type distributions, depending on the number of cumulated shocks. These interarrival times decrease with time, this is modeled by a geometric process, these processes have shown their utility modeling the ageing of the systems and imperfect repair (see [2,3]). The shocks can be fatal or not.…”