Stochastic modeling of quality of systems operating in a heterogeneous environment

We consider systems that are subject to an external mixed Poisson shock process. Each shock can result in a failure of a system with a given probability and is survived with the complementary probability. Each shock additionally decreases the quality function that describes the performance of a system, thus forming the corresponding stochastic process. Expectations (unconditional and conditional on survival) and relevant variability characteristics for the stochastic quality function are derived. Some monotonicity properties of the conditional quality function are investigated and the future values of this function are derived.