An optimal time to sleep for an auto-sleep system

This paper considers an auto-sleep system, which is a kind of intermittently-used system and which goes to sleep if no call for the system occurs for a prespecified time. Such a system can be seen when the system aims to save energy, e.g. electric power. This study provides two models for determining an adequate time for an auto-sleep system to go to sleep. Model 1 formulates the expected energy consumed per unit time to be minimized, while Model 2 defines the energy-efficiency as an objective function to be maximized. For each model, the existence of an optimal time to sleep is examined under a general call distribution to show that there always exists a unique finite optimal time if the distribution has a decreasing failure rates. The results for a Weibull call distribution are also provided for each model along with numerical examples.