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We consider the system efficiency measure as the average asymptotic gain earned by the system for one time unit. Such defined efficiency is a continuous function of the load that the system is to support. Each increment of the load causes an increment of the gain. On the other hand, it also causes a loss of system reliability, giving in this way a loss. This loss follows from the fact that a less reliable system is in the off state more frequently than in the lower-load case. The goal is to find the load that maximizes the efficiency.The lifetime distribution of the system is defined as exponential. As models of the relationship between the load and the gain as well as the relationship between the load and the failure rate, 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 increases are presented.
Jerzy K. Filus (Tue,) studied this question.