Stochastics and Statistics
Extended block replacement policy with shock models and used items

https://doi.org/10.1016/S0377-2217(01)00224-7Get rights and content

Abstract

This paper considers an extended block replacement policy with shock models and used items. 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 1 failure (minor failure) is removed by a minimal repair, whereas type 2 failure (catastrophic failure) is removed by an unplanned (or unscheduled) replacement. After a replacement the shock process resets at 0. Under such a policy, an operating system is preventively replaced by new ones at times iT(i=1,2,…) independently of its failure history. If system fails in ((i−1)T,iTδ) it is either replaced by a new one or minimally repaired, and if in [iTδ,iT) it is either replaced by a used one or minimally repaired. The choice of these two possible actions is based on some random mechanism which depends on the number of shocks since the last replacement. The expected cost rate is obtained, using the results of the renewal reward theory. Various special cases are considered. Our results are shown to extend many of the well-known results for the block replacement policies.

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 the reliability theory is the study of various maintenance policies in order to reduce the operating cost and the risk of a catastrophic breakdown. A well-known preventive maintenance policy is the block replacement policy (see [6], for example). In such a policy, an operating system is preventively replaced by a new one at time iT(i=1,2,…) and at failure. This policy is commonly used when there are a large number of similar systems in service. The main drawback of the block replacement policy is that it is rather wasteful because sometimes almost-new systems are also replaced at planned replacement times. To overcome this undesirable feature, various modifications have been advocated to reduce the wastage (see [3], [4], [5], [7], [8], [9], [11], [13], [14], [15], [17], [20], [22], [24], [25], [26], for example). In particular, the extended block replacement policy with used systems, suggested by Tango [25], is as follows:

  • 1.

    Systems are replaced by new ones at time iT, i=1,2,…

  • 2.

    If systems fail in [(i−1)T,iTδ), they are replaced by new systems and, if they fail in [iTδ,iT), they are replaced by used systems of age T, where 0⩽δT.

Tango's policy creates used systems of age varying randomly from δ to T. However, it uses only used systems of age T and discards used system of age less than T. Murthy and Nguyen [14] extend Tango's policy to the case where failed systems in [iTδ,iT), i=1,2,…, are replaced by used systems with age varying from δ to T, as opposed to replacement by used systems of age T only.

Boland and Proschan [12] consider a periodic replacement of the system subject to shocks and give sufficient conditions for the 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. [10] establish similar results assuming that cost structure is time-dependent. Abdel Hameed [1] shows via a sample path argument that the results of Boland and Proschan [12] and Block et al. [10] hold for any counting process whose jump size is of one unit magnitude. For previous work of a similar shock model, the reader may also refer to Abdel Hameed and Proschan [2], Puri and Singh [18], Sheu [24], and Sheu and Griffith [21], [23].

In this paper an extended block replacement policy with shock models and used items is proposed and analyzed. We extend Tango's policy to the case where if system fails in ((i−1)T,iTδ), it is either replaced by a new one or minimally repaired, and if in [iTδ,iT) it is either replaced by a used one with age varying from δ to T or minimally repaired. The decision to repair or replace the system at failure depends on the number of shocks suffered since the last replacement. The preventive maintenance policy proposed in this paper is defined in Section 2 and its formulation and analysis are given in Section 3. In Section 4 special cases are dealt with.

Section snippets

General model

We consider an extended block replacement policy with shock models and used items according to the following scheme:

  • 1.

    Preventive replacements with new systems are made at times iT, i=1,2,…, at a cost R3, independently of the system's failure history.

  • 2.

    The new system (the used system) is subject to shocks that arrive according to a non-homogeneous Poisson process {N(t);t⩾0} ({Ñ(t);t⩾0}) with intensity function r(t) (r̃(t)) and mean value function Λ(t)=∫0tr(s)ds (Λ̃(t)=∫0tr̃(s)ds) where t is the age

Formulation and analysis

If no planned replacements are considered (i.e., T→∞) and δ=0, then the survival function of the time between the successive type 2 failure of the new system is given byH̄(t)=∑k=0P(N(t)=k,M>k)=∑k=0eΛ(t)Λ(t)kk!P̄k,with density h(t)=−dH̄(t)/dt given byh(t)=∑k=0eΛ(t)Λ(t)kk!r(t)pk+1.

If no planned replacements are considered (i.e., T→∞) and δ=T, then the survival function of the time between the successive type 2 failure of the used system is given byH̃̄(t)=∑k=0P(Ñ(t)=k,M̃>k)=∑k=0eΛ̃(t)Λ̃(t)

Special cases


Case 1. P̄0=1; P̄k=0, k=1,2,…; P̃̄0=1; P̃̄k=0, k=1,2,…

This is the case considered by Murthy and Nguyen [14]. Letting P̄0=1, P̃̄0=1, q1=P̄1/P̄0=0, q̃1=P̃̄1/P̃̄0=0, P̄k=0, P̃̄k=0, k=1,2,…, in (20), we obtainJ(T,δ)=1TR3+R1V(T−δ)+R2T−δT(1+Ṽ(T−t))f(t)dt+R20T−δT−δ−yT−y(1+Ṽ(T−y−t))f(t)v(y)dtdy,where V(t) is the renewal function associated with survival function H̄(t)=exp{−Λ(t)} and Ṽ(t) is the renewal function associated with survival function H̃̄(t)=exp{−Λ̃(t)}.
Case 2. P̄0=1; P̄k=0, k=1,2,…; P

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which greatly enhanced the clarity of the paper. Their suggestions were incorporated directly in the text. This research was supported by the National Science Council of Taiwan, ROC, under Grant No. NSC 88-2118-M-011-001.

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