Discrete and Continuous Operational Calculus in N-Critical Shocks Reliability Systems with Aging under Delayed Information

We study a reliability system subject to occasional random shocks of random magnitudes W 0 , W 1 , W 2 , … occurring at times τ 0 , τ 1 , τ 2 , … . Any such shock is harmless or critical dependent on W k ≤ H or W k > H , given a fixed threshold H . It takes a total of N critical shocks to knock the system down. In addition, the system ages in accordance with a monotone increasing continuous function δ , so that when δ T crosses some sustainability threshold D at time T , the system becomes essentially inoperational. However, it can still function for a while undetected. The most common way to do the checking is at one of the moments τ 1 , τ 2 , … when the shocks are registered. Thus, if crossing of D by δ occurs at time T ∈ τ k , τ k + 1 , only at time τ k + 1 , can one identify the system’s failure. The age-related failure is detected with some random delay. The objective is to predict when the system fails, through the N th critical shock or by the observed aging moment, whichever of the two events comes first. We use and embellish tools of discrete and continuous operational calculus ( D -operator and Laplace–Carson transform), combined with first-passage time analysis of random walk processes, to arrive at fully explicit functionals of joint distributions for the observed lifetime of the system and cumulative damage to the system. We discuss various special cases and modifications including the assumption that D is random (and so is T ). A number of examples and numerically drawn figures demonstrate the analytic tractability of the results.