We consider a mobile connected to a base station, saving energy by shutting off its transceiver, and we try to answer the fundamental question: what is the optimal sleep policy? Firstly, we study the model from optimal control perspective. We consider off-times (periods of inactivity) of unknown duration. We study the question of scheduling ''waking up'' instants in which the mobile communicates with the base station and checks whether the inactivity period is over. There is a cost proportional to the delay from the moment the off-time ends until the mobile discovers it, a (small) running cost while the mobile is sleeping and also a cost for waking up. We show that constant sleep periods are optimal for Poisson arrivals and derive the optimal period. We show that this structure does not hold for other off-time distributions but manage to obtain suboptimal solutions which perform strictly better than the constant one. We finally obtain structural properties for optimal policies for the case of arbitrary distribution of off-times. Technological restrictions often permit a limited set of policies to be implemented. Motivated by this, we investigate classes of policies with specific constraints. What is the optimal policy within each class and what are the optimal parameters for it? To answer these questions, we adopt the parametric optimization approach which entails cost minimization for a given parametrized policy and selection of the best policy among a class. We provide the optimal solution for each class which can be used in closed form or evaluated numerically depending on the case. Our framework allows us to compare the performance of obtained optimal policies, proposed suboptimal policies as well as that of standard policies like IEEE 802.16e.