Replacement policy in a system under shocks following a Markovian arrival process

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Abstract

We present a system subject to shocks that arrive following a Markovian arrival process. The system is minimally repaired. It is replaced when a certain number of shocks arrive. A general model where the replacements are governed by a discrete phase-type distribution is studied. For this system, the Markov process governing the system is constructed, and the interarrival times between replacements and the number of replacements are calculated. A special case of this system is when it can stand a prefixed number of shocks. For this new system, the same performance measures are calculated. The systems are considered in transient and stationary regime.

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(D0,D1). The initial vector is denoted by α.

Assumption 2

The number of minimal repairs before the replacement follows a PHd(γ,L) distribution. We denote by γ0 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. K=3). We assign the following values to the MAP: α=(1,0);D0=-210-3;D1=1030.

Applying the calculations of Section 4 we obtain the stationary probability vector:π0=(0.1875,0625),(π1,π2,π3)=(0.1875,0625;0.1875,0625;0.1875,0625).

It results that the subvectors πi=(0.1875,0625), i=0,1,2,3 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.

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