A reliability semi‐Markov model involving geometric processes

Abstract

SUMMARYWe consider a semi-Markov process that models the repair and maintenance of a repairable system in steady state. The operating and repair times are independent random variables with general distributions. Failures can be caused by an external source or by an internal source. Some failures are repairable and others are not. After a repairable failure, the system is not as good as new and our model reflects that. At a non-repairable failure, the system is replaced by a new one. We assume that external failures occur according to a Poisson process. Moreover, there is an upper limit N of repairs, it is replaced by a new system at the next failure, regardless of its type. Operational and repair times are affected by multiplicative rates, so they follow geometric processes. For this system, the stationary distribution and performance measures as well as the availability and the rate of occurrence of different types of failures in stationary state are calculated.

Some Limit Theorems in Geometric Processes

“…Thereafter, Lam [7] and Lam, Zhang [9] also applied a GP model to the reliability analysis for two-component parallel and series system respectively. For more applications, see [10,11]. On the other hand, Lam [6] and Lam, Chan [8] used GP in data analysis from a series of events with a monotone trend, the theoretical and numerical results both show that GP is a good model for fitting such kind of data.…”

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

“…The device undergoes wear with time, and it can fail because of it. The wear of the device is incorporated considering that between consecutive shocks the successive interarrival times are affected by a degradation factor, in such a way that the wear mechanism follows a geometric process [12]. We must add new assumptions related to the wear.…”

A Monotone Process Maintenance Model for a Multistate System

“…Furthermore, Lam (1992a), Lam and Chan (1998), and Lam et al (2004) applied the GP model to the analysis of data from a series of events. For further reference, see Leung and Lee (1998), Sheu (1999), andPérez-Ocón andTorres-Castro (2002). Moreover, see Lam (2005) for a brief review and more references.…”