Stochastics and Statistics
An extended optimal replacement model of systems subject to shocks

https://doi.org/10.1016/j.ejor.2005.04.042Get rights and content

Abstract

A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur a system has two types of failures: type I failure (minor failure) is rectified by a minimal repair, whereas type II failure (catastrophic failure) is removed by replacement. The probability of a type II failure is permitted to depend on the number of shocks since the last replacement. This paper proposes a generalized replacement policy where a system is replaced at the nth type I failure or first type II failure or at age T, whichever occurs first. The cost of the minimal repair of the system at age t depends on the random part C(t) and deterministic paper c(t). The expected cost rate is obtained. The optimal n and optimal T which would minimize the cost rate are derived and discussed. Various special cases are considered and detailed.

Introduction

It is of great importance to avoid the failure of a system during actual operation when such an event is costly and/or dangerous. In such situations, one important area of interest in reliability theory is the study of various maintenance policies in order to reduce the operating cost and the risk of a catastrophic breakdown. Consider a system that is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement policy for the system.

Barlow and Proschan [5] discussed the traditional approach of replacing a system at failure or at age T, whichever comes first. Boland and Proschan [10] considered periodic replacement of the system and give sufficient conditions for existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. [7] established similar results assuming that the cost structure is time-dependent. Abdel-Hameed [1] shows via a sample path argument that the results of Boland and Proschan [10] and Block et al. [7] hold for any counting process whose jump size is of one unit of magnitude. For previous work of a similar shock model, we refer readers to Abdel-Hammed and Proschan [2], Esary [12], Taylor [33], Feldman [13], Abdel-Hammed and Shimi [3], Zuckerman [35], [36], Brown and Proschan [8], Block et al. [6], Puri and Singh [24], Savits [26], Sheu and Griffith [30], [31] and Sheu [29].

Makabe and Morimura [14], [15], [16] proposed a new replacement model where the system is replaced at the nth failure, and discussed the optimum policy. This model has been generalized by Morimura [17], Park [22], [23], Nakagawa [18], [19], and Nakagawa and Kowada [20]. Park [22] proved that for a Weibull distribution which is widely used for failures, the optimum solution is more cost effective as compared to Barlow and Hunter’s policy. Sheu [27], [28], [29], Sheu and Griffith [31] and Sheu et al. [32] extended the model considered by Nakagawa [18] by introducing general random repair costs and age-dependent minimal repairs.

In this article a generalized replacement model where a system is replaced at age T or at nth type I failure (minor) or first type II failure (major), whichever occurs first, is presented which incorporates minimal repair, replacement, and general random repair costs. The models with two types of failures and threshold replacement policy with operating and non-zero repair phase type distributions was considered in Neuts et al. [21] This model in this paper is a generalization of the model “Policy IV” called by Morimura [17] and it can be considered as a generalization on several previously known policies for the class of systems in which the minor failure results in a minimal repair and the non-homogeneous Poisson process is used in modeling the failure process. We also assume that repairs and replacements can be done instantly. Precisely speaking, this assumption is not true in real situations. However, when these repair or replacement times are relatively small compared to the operational times, these non-operational times can be negligible. Our model is also motivated by the maintenance problem of some practical systems. For example, in one of the top 100 most wired universities in the USA ranked by Yahoo, every lecture hall has an expensive multi-media networked computer system. It is critical to maintain this system highly reliable as a system failure during a lecture can create a huge frustration to both professors and a large number of students. The system is quite expensive to replace so that a replacement occurs only when either a major non-repairable failure due to several reasons including vandalism occurs or when the system has been used for a maximum period of time, T. The system is also subject to some repairable non-major failures. If a certain number of repairable failures reaches a threshold value, n, before T, the system is also replaced because the high frequency of minor failures indicates a low reliability and may cause too many interruptions in classes. Note that in this situation, the repair times or the replacement times are negligible compared to the operational times. The model presented in this paper can help instructional media service people find an optimal maintenance policy for these expensive systems installed in all large lecture halls. To achieve this objective, we derive the expected cost rate for this model. The optimal number n and optimal age T which would minimize the cost rate under certain conditions are determined. Various results are obtained as special cases of our model such as Barlow and Hunter [4], Makabe and Morimura [14], [15], [16], Morimura [17], Tilquin and Cléroux [34], Cléroux [11], Park [22], [23], Nakagawa [18], and Boland [9].

The rest of this paper is organized as follows: The replacement model proposed in this paper is described in Section 2 and its formulation and analysis are given in Section 3. The total expected long-run cost per unit time is found and Theorem 1 gives general optimization results. Various special cases and some concluding remarks are given in Section 4.

Section snippets

General model

We consider a replacement model of a system subject to shocks in which minimal repair or replacement take place according to the following scheme:

  • 1.

    A system is subject to shocks that arrive according to a non-homogeneous Poisson process {N(t); t  0} with intensity function r(t) and mean value function Λ(t)=0tr(u)du where t is the age of the system. We assume that r(t) is a continuous and positive increasing function for t  0.

  • 2.

    As shocks occur the system has two types of failures: type I (minor)

Formulation

If T = ∞, the survival function of the time between successive replacements is given byG¯(t)=k=0n-1P(N(t)=k,M>k)=k=0n-1e-Λ(t)Λ(t)kk!P¯k,with density g(t)=-(dG¯(t)/dt) given byg(t)=k=0n-1e-Λ(t)Λ(t)kk!r(t)P¯k-k=0n-2e-Λ(t)Λ(t)kk!r(t)P¯k+1.

Let random variable Y represent the time until nth type I failure or first type II failure for our model with T = ∞. That is, Y be the random variable with survival function G¯(t), and let rG(t)=g(t)/G¯(t) be the hazard rate function of G. ThenrG(t)=g(t)G¯(t)=[1-τ

Special cases and concluding remarks

In this section, we demonstrate that several previous models are special cases of our model.
Case 1. T = ∞, P¯k=1 for k = 0, 1, 2,  , and g(C(t), c(t)) = c1.

Makabe and Morimura [14], [15], [16] considered this case in which only minimal repairs with fixed cost c1 are performed on failure before the nth failure, whereas a system is replaced at the nth failure. In this case, if we put T = ∞, P¯k=1 for k = 0, 1, 2, …), and h(t) = EC(t)[g(C(t), c(t))] = c1 in (10), thenB(n,;{1})=R1+(n-1)c1k=0n-10e-Λ(t)Λ(t)kk!dt,n=1,2,

Acknowledgements

We are very grateful to the referees for their insightful comments and suggestions which improved the paper significantly.

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