A cyclic renewal process is considered as an extension of an alternating renewal process, whereeach of the underlying independently and identically distributed (i.i.d.) nonnegative random increments iscomposed of multiple stages. Such a process may be appropriate for analyzing optimal preventive maintenancepolicies for production management, where a pair of two stages representing an uptime until a minorfailure and the subsequent minimal repair time would be repeated until it is decided to conduct a completeoverhaul. In order to address economic problems in such applications, we also introduce a reward processwith jumps defined on the cyclic renewal process. When the system is running in stage j, the profit growslinearly at the rate of ρ(j). Upon a minor failure, the subsequent minimal repair in stage (j + 1) incurs thelinear cost at the rate of ρ(j + 1). In addition, the fixed cost may be imposed whenever either a minimalrepair or a complete overhaul takes place, resulting in jumps of the reward process. The problem is thento 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 processis 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 productionmanagement.
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