Bayesian computation for geometric process in maintenance problems

Abstract

Geometric process modeling is a useful tool to study repairable deteriorating systems in maintenance problems. This model has been used in a variety of situations such as the determination of the optimal replacement policy and the optimal inspection-repair-replacement policy for standby systems, and the analysis of data with trend. In this article, Bayesian inference for the geometric process with several popular life distributions, for instance, the exponential distribution and the lognormal distribution, are studied. The Gibbs sampler and the Metropolis algorithm are used to compute the Bayes estimators of the parameters in the geometric process. Simulation results are presented to illustrate the use of our procedures.

Introduction

In the early studies of the maintenance problems, previous results on repair replacement models usually base on the assumption that a failure system after repair will be ‘as good as new’ and the repair times can be neglected. The successive operating times thus generate a renewal process. These models can be classified as perfect repair model. However, for repairable deteriorating systems, the problem is different from that described above. For example, in machine maintenance, in view of ageing and accumulated wear, the operating times of a machine after repair will become shorter and shorter and the repair times will become longer and longer. Thus, an appropriate model should be as follows: the operating times are stochastically decreasing and the consecutive repair times are stochastically increasing. Stadje and Zuckerman [20] developed a more general repair replacement model by introducing a monotone process. Lam [9], [10], [14] introduced a special monotone process which was called the geometric process. This process was applied to a replacement model in which the operating times of system form a non-increasing geometric process and consecutive repair times constitute a non-decreasing geometric process.

Let us define the geometric process (see Lam [9], [10], [14]). Suppose that X₁, X₂,  …, is a sequence of random variables. If there exists a > 0 such that {aⁱ⁻¹X_i, i = 1, 2,  … } forms a renewal process (RP), then {X₁, X₂,  … } is called a geometric process (GP) and the real number a is called the ratio of the GP.

It has been showed that a GP is stochastically non-increasing if a ≥ 1 and stochastically non-decreasing if 0 < a ≤ 1. If a = 1, it reduces to an RP. Therefore, the geometric process is an extension of the renewal process and provides us with some nice features. For example, we assume that {X₁, X₂,  … } is a GP with ratio a and X₁ has a density function f(t) with the mean λ and variance σ². Then X_i have a density function aⁱ⁻¹f(aⁱ⁻¹t) with E(X_i) = λ / aⁱ⁻¹ and Var(X_i) = σ² / a²⁽ⁱ⁻¹⁾, (i = 1, 2,  … ). The geometric process has important applications in many fields of maintenance problems. Lam [8], [10] and Zhang [22] used the geometric process to determine respectively the N and (T, N) mixing optimal replacement policy for repairable deteriorating systems under the long-run average cost per unit time. Lam [11] developed the optimal inspection-repair-replacement (IRR) policy in standby systems using GP, and Lam and Zhang [16] explored the analysis of a two-unit series with a geometric process model.

As the geometric process is an important technique for studying repairable deteriorating systems in maintenance problems, it is worth to note that the geometric process has three crucial parameters a, λ and σ². It is necessary to estimate these parameters in the application of the geometric process to the optimal replacement problem. Our main objective in this article is to estimate the unknown parameters for a GP {X₁, X₂,  … } based on the Bayesian framework.

The main computational tool to accomplish the above objective is the Gibbs sampling algorithm that has been successfully applied to many statistical problems. The Gibbs sampler (see, e.g., Geman and Geman [6], Gelfand and Smith [4] and Gelman and Rubin [5] is often employed to calculate the minimum mean squared error estimates of the unknowns. It uses random draws from the conditional distributions of each component of random vector given all the other components. The sampler only requires the ability to draw random samples from the conditional distributions of the parameters involved. When the conditional distributions is not simple, the Metropolis algorithm is used. The Metropolis algorithm (see, e.g., Tierney [21] and Siddhartha and Edward [19]) is a useful and straightforward device to sample from a complicated distribution approximately. In the following sections, we see that all unknown parameters in the geometric process are estimated via a combination of the Gibbs sampler and Metropolis algorithm.

On the other hand, in modeling a set of data from a point process with trend, a common method is to apply a nonhomogenous Poisson process with a monotone rate (see Ascher and Feingold [1]). Lam [12] used geometric process to model a point process with trend. He also studied a modified moment estimation (MME) of parameters for GP. Asymptotic normal properties of the MME for GP have been studied by Lam [12]. Lam and Chan [17] and Lam et al. [18] developed asymptotic results for maximum likelihood estimator (MLE) for GP with lognormal distribution. It had been shown that the MLE is more efficient than the MME.

This article is organized as follows. Section 2 introduces the GP maintenance problem and some statistical inference in fitting trend data. In Section 3, we provide a Bayes approach to the estimation of the unknown parameters and develop the posterior distributions for several special GP using the Gibbs sampler. We also exhibit details on how to implement the Gibbs algorithm and the Metropolis algorithm for generating samples from the posterior distribution. Section 4 develops the Bayes inference for the mean parameter λ, variance parameter σ² and the future time between failures. Simulation results are examined in Section 5. Concluding remarks are presented in Section 6.