## Some shock models in reliability theory: a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Statistics at Massey University, Palmerston North, New Zealand

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##### Date

1986

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Massey University

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The Author

##### Abstract

This thesis is concerned with the lifetime distribution of a device subject to environmental shocks. The terms "device" and "shock" are used here in an abstract sense and although industrial interpretations are the most obvious, the models described in this thesis can also be applied in other fields, for example, in biology and finance. Several models are presented and in each case the main question of interest is to determine the class of distributions to which the lifetime distribution associated with the model belongs. This approach to the study of shock models is taken since it is often difficult to derive an explicit expression for the lifetime distribution of a device, but if the class to which the distribution belongs can be identified it is usually possible to obtain a bound on the distribution. Since classes of lifetime distribution have an important role to play in the study of shock models and in reliability theory generally, the first part of this thesis is devoted to a review of the classes which have proved useful in these areas. The classes are defined and some justification for their use in reliability theory is provided. In addition, alternative characterisations of the classes are given and it is shown that a function which is closely related to the Laplace transform can be used to characterise all the classes. The discussion of shock models commences, in chapter two, with a survey of results pertaining to the standard shock model:- where H̅(t) = 1 - H(t) is the lifetime distribution of a device subject to shocks whose arrival is governed by the stochastic counting process {N(t)}. The probability that the device survives K shocks is given by P̅к where k=0,1,2,...... This model has received a good deal of attention in the literature (see, for example, Esary, Marshall and Proschan (1973), A-Hameed and Proschan (1975), Klefsjo (1981, 1985). The most striking feature of the model is that under appropriate conditions on {N(t)} the lifetime distribution inherits its class from the discrete class of the survival probabilities. Results are presented under a variety of assumptions on {N(t)} ranging from the assumption of a homogeneous Poisson process to the assumption that {N(t)} is a generalised renewal process. In addition, a model where {N(t)} is assumed to be a doubly stochastic Poisson process is introduced. For the more general models it is often the case that the life distribution H inherits its class not only from the survival probabilities but also from the class of the shock interarrival time distribution. The final part of this thesis is concerned with shock models in which failure occurs according to some specified mechanism. In particular, two methods of failure are considered. Firstly, the case where failure occurs on the occurrence of a shock which exceeds some critical threshold is studied and, secondly, the case where failure occurs when the total accumulated damage due to shocks exceeds some critical threshold is considered. In both cases, the initial approach is to use the standard model with an appropriate structure imposed on the survival probabilities A more general approach which allows for some dependency between the shock magnitudes and shock interarrival times and, in the case of the cumulative damage model, for wear or recovery between shocks, is then adopted. Such models have been studied by Shanthikumar and Sunmita (1983, 1984) and by Shanthikumar (1984). Their results are summarised and the importance of the class of the shock inter-arrival time distributions in determining the class to which the lifetime distribution of the model belongs is noted. In addition, some minor extensions to Shanthikumar's (1984) work on the cumulative damage model are made.

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##### Keywords

Reliability (Engineering), Shock