Optimum Replacement of a System Subject to Shocks: A Mathematical Lemma

We consider a system that is subject to shocks that occur randomly over time. We assume that the system must be replaced after it has functioned for a random length of time τ, a moment of a major failure that is a stopping time with respect to the process {N(t), t ≥ 0} that models the number of shocks occurring in [0, t]. It may, however, be economical to replace the system at time min(t, τ) prior to its failure, for some fixed but optimally chosen t. Costs are due to shocks, maintenance and replacement. We introduce an optimality criterion, based on cost considerations, which we use to find an optimal replacement time t₀. In the process, we prove a general mathematical lemma that helps in arriving at the optimal replacement time for general classes of processes {N(t)}, stopping times τ and the cost structures involved. We also give a discrete time version of the lemma.