Analysis of a single-facility stochastic maintenance system with a constant lead time

Abstract

In this thesis, we present a continuous review (O,R) replacement system for a single-facility stochastic maintenance system with a constant lead time, where the inter-shock arrival times and the magnitudes of damages caused by shocks are assumed to be two different arbitrary stochastic processes. In the replacement system, the shock position process {$X_t; t≥0$} totally depends upon the cummulative damage quantity of shocks, represented by a shock process {S(t); t≥0}. A new facility for replacement is ordered whenever the shock position exceeds the replacement level R, where each procurement lead time is a positive constant. To analyze such replacement system, it is, first, shown that the shock positions and the cummulative damages during the procurement lead time are independent in stationary. Upon the result, the stationary distribution of on-hand shock levels is derived. Finally, a long-run expected average cost function is developed, with which examples for computing the optimal replacement level $R^*$ are treated when the shock process follows a compound poisson process.