Predictive analyses for nonhomogeneous Poisson processes with power law using Bayesian approach

Abstract

Nonhomogeneous Poisson process (NHPP) also known as Weibull process with power law, has been widely used in modeling hardware reliability growth and detecting software failures. Although statistical inferences on the Weibull process have been studied extensively by various authors, relevant discussions on predictive analysis are scattered in the literature. It is well known that the predictive analysis is very useful for determining when to terminate the development testing process. This paper presents some results about predictive analyses for Weibull processes. Motivated by the demand on developing complex high-cost and high-reliability systems (e.g., weapon systems, aircraft generators, jet engines), we address several issues in single-sample and two-sample prediction associated closely with development testing program. Bayesian approaches based on noninformative prior are adopted to develop explicit solutions to these problems. We will apply our methodologies to two real examples from a radar system development and an electronics system development.

Introduction

Several reliability models have been proposed in the literature for estimating system reliability during development testing and evaluating software reliability. Amongst, nonhomogeneous Poisson process (NHPP) with intensity function

$${\lambda }_t\widehat{=}\lambda (t)=(\beta /\alpha )(t/\alpha {)}^{\beta -1},\alpha ,\beta >0\text{,}$$

is perhaps the most widely used model among governments and industries. In the literature, this NHPP possesses different names such as Weibull process, power-law process and Duane model (Duane, 1964).

Applications of the Weibull process on assessing hardware reliability growth and detecting software failures were investigated extensively by various authors. For instance, classical (or frequentist) estimations on unknown parameters, mean time between failures and the current system reliability were studied by Crow, 1974, Crow, 1982, Bain (1978), Engelhardt and Bain (1978), Lee and Lee (1978), Bain and Engelhardt (1980), Joe (1989) and Huang et al. (2003) while those based on traditional Bayesian inferences were investigated by Kyparisis and Singpurwalla (1985), Guida et al. (1989), Bar-Lev et al. (1992) and Kuo and Yang (1995). Using Gibbs sampling, Kuo and Yang (1996) presented a unified approach for analyzing more general NHPPs which include several models as special cases (e.g., the Weibull process, the Cox–Lewis (1966) model with intensity $\lambda (t)={\mathrm{e}}^{\alpha +\beta t}$, the Goel–Okumoto (1979) model with $\lambda (t)=\alpha \beta {\mathrm{e}}^{-\beta t}$, the delayed S-shaped model (Yamada et al., 1983) with $\lambda (t)=\alpha {\beta }^{2}t{\mathrm{e}}^{-\beta t}$, and the Musa–Okumoto (1984) model with $\lambda (t)=\alpha /(t+\beta )$). By using Markov chain Monte Carlo method, Ryan (2003) presented Bayesian inferences for NHPPs with a variety of more flexible families of intensities built on the notion of switching in time between two simple constituent intensities. For more detailed discussions, one can refer to Xie (1991), Lyu (1996), Singpurwalla and Wilson (1999) and Verma and Kapur (2006).

As noted by Meth (1992, p. 339), “reliability growth models can potentially provide the basis for (i) planning reliability growth tests, (ii) monitoring progress and estimating current reliability, and (iii) forecasting future reliability improvements.” In other words, reliability growth model is a powerful tool for forecasting and prediction (see, e.g., Aitchison and Dunsmore, 1975, West and Harrison, 1989, Geisser, 1993). In particular, predictive analyses are useful for determining when to terminate a development process. Usually, a prediction interval is constructed to indicate the time frame when the kth $(k>0)$ future observation will occur with pre-determined confidence level. However, literature for predictive inference on Weibull process is relatively scattered. For instance, Lee and Lee (1978) proposed a numerical integration method for constructing prediction limits for the kth $(k>0)$ future observation in frequentist settings. Engelhardt and Bain (1978) derived a comparable result which does not involve numerical integration. Alternatively, Tian (1992) presented a closed-form Bayesian prediction interval for more general observed data. In addition, Pulcini (2001) studied the Bayesian prediction of the future failure times and the number of failures in a future time interval for a repairable system subject to minimal repairs and periodic overhauls. Sen (2002) investigated Bayesian prediction of the Weibull intensity. Pfefferman and Cernuschi-Frias (2002) provided a nonparametric prediction procedure and Pievatolo et al. (2003) presented an application of predicting the expected number of failures in underground trains during a given period.

This article focuses on single-sample and two-sample predictive inferences for the Weibull process via Bayesian approach. Driven by the demand in developing complex high-cost and high-reliability systems (e.g., weapon systems, aircraft generators, jet engines), we first identify four issues in single-sample prediction and three issues in two-sample prediction associated closely with development testing program and derive the corresponding predictive distributions in Section 2. The main results for single- and two-sample predictions are presented in Sections 3 and 4, respectively. Two real examples from a radar system development and an electronics system development are used to illustrate the proposed methodologies in Section 5. We conclude with a discussion in Section 6 and put the mathematical proofs in the Appendix.