Reliability systems subject to non-fatal shocks

The reliability modeling of numerous physical systems is critical in the prevention of system failure. In many instances, the failure rate of system components is a function of non-fatal shocks or stresses to the system that occur at discrete points in time. These shocks are assumed to be identical and reparable, and they impact the failure rate of the system in a non-linear fashion via the cumulative and current number of shocks. In this paper, we demonstrate the application of the cumulant derivation procedure to this reliability system in a Markovian environment. This approach utilizes a truncated cumulant generating function to generate a set of ordinary differential equations whose numerical solution approximates the reliability function. These approximations are obtained under various truncation levels whereby this approach is shown to be tractable for large systems.