Dynamic analysis of a reward process defined on a cyclic renewal process with applications to preventive maintenance problems

A cyclic renewal process is considered as an extension of an alternating renewal process, where each of the underlying independently and identically distributed (i.i.d.) nonnegative random increments is composed of multiple stages. Such a process may be appropriate for analyzing optimal preventive maintenance policies for production management, where a pair of two stages representing an uptime until a minor failure and the subsequent minimal repair time would be repeated until it is decided to conduct a complete overhaul. In order to address economic problems in such applications, we also introduce a reward process with jumps defined on the cyclic renewal process. When the system is running in stage j, the profit grows linearly at the rate of ρ(j). Upon a minor failure, the subsequent minimal repair in stage (j+1) incurs the linear cost at the rate of ρ(j+1). In addition, the fixed cost may be imposed whenever either a minimal repair or a complete overhaul takes place, resulting in jumps of the reward process. The problem is then to determine when to conduct a complete overhaul so as to maximize the total reward in the time interval (0,T]. A multivariate Markov process generated from both the cyclic renewal process and the reward process is studied extensively, yielding various new transform results explicitly and deriving their asymptotic expansions. These results are used to numerically explore optimal preventive maintenance policies for production management.