Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method

This paper investigates the M/M/R machine repair problem with second optional repair. Failure times of the operating machines are assumed to be exponentially distributed with parameter $\lambda $. Repair times of the first essential repair and the second optional repair are assumed to follow exponential distributions. A failed machine may leave the system either after the first essential repair with probability $(1-\theta)$, or select to repair for second optional repair with probability $\theta$ $(0 \le \theta \le 1)$ at the completion of the first essential repair. We obtain the steady-state solutions through matrix-analytic method. A cost model is derived to determine the optimal number of the repairmen, the optimal values of the first essential repair rate, and the second optional repair rate while maintaining the system availability at a specified level. We use the direct search method to deal with the number of repairmen problem and the Newton-Quasi method for the repair rate problem to minimize the system operating cost while all the constraints are satisfied.