A Versatile Stochastic Maintenance Model with Reserve and Super-Reserve Machines

Abstract

This article analyzes the maintenance of production system with unreliable machines. This system includes a repair facility and three types of “unreliable” machines: the main facility of working and reserve machines, and an auxiliary facility of “super-reserve” machines. Operating times of working machines are exponentially distributed. Upon failure, a working machine is immediately replaced by reserve machines available. Defective machines line up for repair, whose duration is arbitrarily distributed. Refurbished machines return to the main facility. If the main facility is restored to its original quantity (i.e., all machines are intact), the repair facility leaves on routine maintenance; all w+1 reserve machines are temporarily blocked and renewals come from the super-reserve group until the latter becomes exhausted. Then, the busy period is regenerated. The techniques include two-variate Markov and semi-regenerative processes, and a duality principle, to find the probability distribution of the number of intact machines. Explicit formulas obtained demonstrate a relatively effortless use of functionals of the main stochastic characteristics (such as expenses due to repair, maintenance, waiting, and rewards for higher reliability) and optimization of their objective function. Applications of such models include computer networking, human resources, and manufacturing processes.