An optimal time to sleep for an auto-sleep system considering multi-usage states

An auto-sleep system is defined by the following two properties: 1. call for the system occurs randomly and intermittently he system automatically goes to sleep if there occurs no call during a prespecified time T. nsiders four states: 1. eep rm-up nusage age. uch a system, the time to sleep has been discussed based on suitable criteria. This study extends the model for an auto-sleep system so that the model can deal with multi-usage states. With a view to determining an optimal time to sleep under the extended model, the expected energy consumed per unit time is formulated as a criterion to be minimized. The existence of an optimal time to sleep is examined under a general call distribution. Numerical examples are also provided for a Weibull as well as a log-normal call distribution.