Failure Distributions Associated With General Compound Renewal Damage Processes

S. Zacks1

Department of Mathematical Sciences Binghamton University [email protected]

Key words: Failure Distributions, Reliability, Hazard Functions, Compound Renewal, Damage Processes

1 Introduction

In a recent paper [Z04] distributions of failure times due to random cumulative damage process were investigated. In particular the previous study was focused on random damage processes driven by non-homogeneous compound Poisson process, with a Weibull intensity function and exponential damage in each occurrence. The present paper generalizes the previous results to damage processes, which are driven by compound renewal processes and general damage distributions. The failure time is the first instant at which the cumulative damage crosses the system's threshold. In Section 2 we present the cumulative damage process as a general compound renewal process. The density function of the associated failure distribution is given, as well as its moments. In Section 3 we consider the special case of a homogeneous compound Poisson damage process (CCDP) with exponentially distributed jumps. The density of failure times, when the intensity A of the CCDP is random (doubly stochastic Poisson process) is derived for A having a Gamma distribution. The hazard function for this doubly stochastic case is illustrated in Figure 2. The results of Section 3 are extended in Section 4 to Erlang damage size distributions. The reader is referred to the book of Bogdanoff and Kozin [BK85] for illustrations of random cumulative damage processes. They used a discrete homogeneous Markov Chain to model the extent of damage and failure times (phase-type distributions). The reader is referred also to the book of Bogdanovicius and Nikulin [BN02], and the papers of Wilson [W00], Kahle and Wendt [KW00], Aalen and Gjessing [AG03].

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