Optimal replacement policies determined using arithmetico-geometric processes

In this paper, the author proposes new repair–replacement models for a deteriorating system, in which the successive operating times of the system form an arithmetico-geometric process and are stochastically non-increasing, while the successive repair times after failure also constitute an arithmetico-geometric process but are stochastically non-decreasing. Two kinds of replacement policy are considered, one based on the working age (a continuous decision variable) of the system and the other determined by the number of failures (a discrete decision variable) of the system. These policies are considered together with the performance measures, namely loss (or its complement profit), cost, and downtime (or its complement availability) respectively. Applying the well-known results of renewal reward processes, the author derives expressions for the long-run expected performance measure per unit time, and for the long-run expected performance measure per unit time of good operating condition, under the two kinds of policy proposed.