Imperfect repair and lifesaving in heterogeneous populations

doi:10.1016/j.ress.2006.09.018

Reliability Engineering & System Safety

Volume 92, Issue 12, December 2007, Pages 1671-1676

https://doi.org/10.1016/j.ress.2006.09.018 Get rights and content

Abstract

In this theoretical paper we generalize the notion of minimal repair to the heterogeneous case, when the lifetime distribution function can be modeled by continuous or a discrete mixture of distributions. The statistical (black box) minimal repair and the minimal repair based on information just before the failure of an object are considered. The corresponding failure (intensity) rate processes are defined and analyzed. Demographic lifesaving model is also considered: each life is saved (cured) with some probability (or equivalently a proportion of individuals who would have died are now resuscitated and given another chance). Those who are saved experience the statistical minimal repair. Both of these models are based on the Poisson or non-homogeneous Poisson processes of underlying events, which allow for considering heterogeneity. We also consider the new model of imperfect repair in the homogeneous case and present generalizations to the heterogeneous setting.

Introduction

Most of the papers on reliability and survival analysis deal with the homogeneous case although one can hardly find homogeneous populations in practice. Heterogeneity can introduce new and sometimes unexpected features in reliability analysis when compared with a homogeneous case. It is well known that the (observed) failure rate in a heterogeneous population does not follow the pattern of the subpopulations failure rates. For instance, the mortality rate of a homogeneous human population is approximately an exponentially increasing function (Gompertz law), but recently the deceleration of this increase for old ages was observed. A natural explanation of this fact lies in the population heterogeneity.

In conventional reliability analysis of repairable systems, it was also always assumed that objects to be repaired ‘are chosen’ from a homogeneous population. It turns out, that generalization to the heterogeneous case is straightforward only for the perfect repair. In this case, the process of functioning of an instantaneously repaired item can be, as usually, modeled by the renewal process with a corresponding mixture as a distribution of inter-arrival times.

Even the simplest type of imperfect repair-a minimal repair, complicates modeling to a great extent, which is shown in 2 Homogeneous case, 3 Heterogeneous minimal repair model. In Section 4, we introduce a new imperfect repair model based on the proportional impact of the repair action on the corresponding failure rate and generalize this approach to the heterogeneous setting. In Section 5 the heterogeneous lifesaving model is considered. This model is also based on the notion of minimal repair and can be considered as the corresponding application.

This is a theoretical note, but we believe that the results modeling an impact of heterogeneity on reliability characteristics of real objects can be important in various applications. They show specifically that this impact should be taken into account in reliability analysis of repairable systems.

Section snippets

Homogeneous case

Consider an object with an absolutely continuous time to failure cumulative density function (Cdf) F(t) and a failure rate λ(t), which starts operating at t=0. Assume that the repair action is performed instantaneously upon failure. The repair is usually qualified as perfect, if the Cdf of the repaired object is F(t) (as good as new) and as the minimal repair at time x, if its Cdf is: $F (t | x) \equiv 1 - \frac{1 - F (t + x)}{1 - F (x)} .$

It is clear that the minimal repair does not change the failure rate of our object.

Heterogeneous minimal repair model

Let T⩾0 be a lifetime random variable (r.v.) with the Cdf F(t). As usually, denote the survival function by $\bar{F} (t)$ . Assume that F(t) is indexed by a r.v. Z: $P (T ⩽ t | Z = z) \equiv P (T ⩽ t | z) = F (t, z)$ and that the probability density function (pdf) f(t,z) exists. Then the corresponding failure rate λ(t,z) is defined by $f (t, z) / \bar{F} (t, z)$ . Let Z be interpreted as a non-negative r.v. with support in [a,b], a⩾0, b⩽∞ and the pdf π(z).

Another meaningful interpretation defines an unobserved Z as a frailty in the

A new model of imperfect repair

Firstly, we define this new model for the conventional homogeneous case. Assume, as previously that the failure rate λ(t) of the governing distribution is monotonically increasing (IFR). Let the first failure occurs at t=t₁. Define the imperfect repair as repair, decreasing the pre-failure value of λ(t) in the following way: $λ (t, t_{1}, k_{1}) = {\begin{matrix} λ (t), & 0 ⩽ t < t_{1}, \\ k_{1} λ (t), & t ⩾ t_{1}, \end{matrix}$ where k₁ is an improvement factor: $\frac{λ (0)}{λ (t_{1}) - λ (0)} ⩽ k_{1} ⩽ 1 .$ The case k₁=1 corresponds to a minimal repair. The subsequent cycles are defined in

Heterogeneous lifesaving model

In this section, we are coming back to the lifesaving model of Section 2. Let θ(t) in Eqs. (5), (6) be constant in time for simplicity: θ(t)=θ and assume that it is a random variable (independent of the non-homogeneous process P_t, t⩾0) with support in [0,1]. This is another way of implementing the population heterogeneity into the model. Indeed, for instance, probability of survival after some diseases can vary a lot in the population of the same age. The same argument holds for repairable

Conclusions

Repeated statistical minimal repair in homogeneous populations gives rise to the non-homogeneous Poisson process of failures (repairs), which is a nice and simple model to deal with. Mixture of distributions is a useful tool for modeling heterogeneity. It is well known that the shape of the failure rate in heterogeneous populations can differ dramatically from the baseline failure rate. On the other hand, it is not trivial to define the analog of minimal repair for heterogeneous populations. A

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Imperfect repair and lifesaving in heterogeneous populations

Abstract

Introduction

Section snippets

Homogeneous case

Heterogeneous minimal repair model

A new model of imperfect repair

Heterogeneous lifesaving model

Conclusions

Change of life distribution via hazard transformation: an inequality with application to minimal repair

Math Oper Res

Optimal replacement under a minimal repair strategy: a general failure model

Adv Appl Prob

Stochastic models in reliability

A general minimal repair model

J Appl Prob

Age-dependent minimal repair

J Appl Prob

Statistical and information based minimal repair for k out of n systems

J Appl Prob