The load optimization of a repairable system with gamma-distributed time-to-failure

We consider a system which supports a load in such a way that the profit earned by the system for one time unit is a continuous and increasing function of the amount of the load. The time-to-failure distribution of the system is assumed to be gamma. Both parameters of the distribution are continuous functions of the load such that the mean time-to-failure is a decreasing function of the load.

The system earns the profit only when it is on. When it is off, it is repaired, and the losses are assumed to be proportional to the repair time. We define the average asymptotic gain earned by the system in a single time unit; this quantity can be thought of as a system efficiency measure.

Assuming that the load is a controllable variable, the goal is to find a value of the load that maximizes the so-defined efficiency. To model the relationship between the load and the parameters of the time-to-failure distribution (gamma) as well as the relation between the load and the profit, power and exponential functions have been taken.

The analytical conditions for the existence of maximum positive gain and the analysis of the behaviour of the gain as the load varies are presented.