On some characteristics of quality for systems operating in a random environment

We consider systems that are operating in a random environment modeled by an external shock process. Performance of a system is characterized by a quality (output) function that is decreasing (due to degradation) in the absence of shocks. The important feature of our model is that shocks affect the failure rate of a system directly, and at the same time, each shock contributes to the additional decrease in the quality function forming the corresponding stochastic process. Expectations (unconditional and conditional on survival) and variability of this process are analyzed. Some monotonicity properties of the conditional quality function are discussed and expressions for the future values of this function are derived