Replacement policy in a system under shocks following a Markovian arrival process
Section snippets
The model
Let a system be subject to shocks. After every shock, it is minimally repaired, and the repair time is negligible. After a series of random shocks, the system is replaced by a new and identical one. The following assumptions establish they operate. Assumption 1 The shocks arrive following an MAP. The initial vector is denoted by . Assumption 2 The number of minimal repairs before the replacement follows a distribution. We denote by the probability that the system is replaced to the first failure.
Under
Replacements
In this Section we show that the replacements arrive following a PH-renewal process. The number of replacements at any time is calculated.
Model with a limited number of shocks
In this section we consider the model given in Section 2, replacing Assumption 2 by a new assumption limiting the number of shocks. Assumption 2bis The system can undergo K shocks before being replaced.
Numerical application
We illustrate the model with a fixed number of shocks before replacements by means of a numerical example. We consider that the system is replaced when it has undergone three minimal repairs (i.e. ). We assign the following values to the MAP:
Applying the calculations of Section 4 we obtain the stationary probability vector:
It results that the subvectors , are identical. The
Conclusions
We consider an arrival process with dependence between the interarrival times, and we study a classic problem under a new methodology that allows us to reach quantities under the Markovian structure. Previous studies of this model or other related ones, can be obtained from this as particular cases. So, when the arrivals follow an homogeneous Poisson process, a finite mixture of homogeneous Poisson processes, or a PH-renewal process. We introduce a limited number of shocks before replacement,
Acknowledgment
The authors are very grateful to three referees who, with their comments, have improved the final version of the paper. This paper is partially supported by the Ministerio de Educación y Ciencia, España, under the Grant MTM2007-61511.
References (13)
State-age-dependent maintenance policies for deteriorating systems with Erlang sojourn time distributions
Reliab Eng Syst Saf
(1997)- et al.
A multiple system governed by a quasi-birth-and-death process
Reliab Eng Syst Saf
(2004) - et al.
A deteriorating system with two repair modes and sojourn times phase-type distributed
Reliab Eng Syst Saf
(2006) - et al.
A multiple warm standby system with operational and repair times following phase-type distributions
Eur J Oper Res
(2006) - et al.
Reliability of a system under two types of failures using a Markovian arrival process
Oper Res Lett
(2006) - et al.
Availability of a system maintained thorough several imperfect repairs before a replacement or a perfect repair
Stat Probab Lett
(2000)
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