Optimal sampling for state change detection with application to the control of sleep mode

This work considers systems with inactivity periods of unknown duration. We study the question of scheduling ¿waking up¿ instants in which a server can check whether the inactivity period is over. There is a cost proportional to the delay from the moment the inactivity period ends until the server discovers it, a (small) running cost while the server is away and also a cost for waking up. As an application to the problem, we consider the energy management in WiMax where inactive mobiles reduce their energy consumption by entering a sleep mode. Various standards exist which impose specific waking-up scheduling policies at wireless devices. We check these and identify optimal policies under various statistical assumptions. We show that periodic fixed vacation durations are optimal and derive the optimal period. We show that this structure does not hold for other inactivity distributions but manage to obtain some suboptimal solutions which perform strictly better than the periodic ones. We finally obtain structural properties for optimal policies for the case of arbitrary distribution of inactivity periods.