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The lifelengths of components of a system are usually dependent due to the common random production and operating environments. In this paper, we introduce a multi-variate pure jump Markov process to describe a large class of damage processes on various system components driven by common environmental shocks, and establish some dependence properties (association) for such a process and its multivariate increment process. These strong association properties describe both spatial dependence and temporal dependence of a multivariate pure jump process, and also provide a vehicle to derive some structural properties of component lifelengths of the systems operating in such an environment. Some bounds for the joint survival functions of component lifelengths are also obtained.

We investigate the nature of the set of emptiness times of a dam whose release rate depends on the content and whose cumulative input process is a pure-jump Lévy process. Detailed results are obtained for stable input processes and release functions of the form r(x) = xβ I(o,∞)(x).

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Stochastic models for dependent life lengths induced by common pure jump shock environments

Emptiness times of a dam with stable input and general release function

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2 results

Stochastic models for dependent life lengths induced by common pure jump shock environments

Emptiness times of a dam with stable input and general release function