Optimal energy-aware control policies for FIFO servers

We study the optimal control of sleep states in an energy-aware M/G/1-FIFO queue. Two families of policies are considered, referred to as randomized and sequential policies. Under a randomized policy, the server upon becoming idle can wait for a random period before entering, potentially randomly, any of the possible sleep states. The server enters setup after a possibly random number of jobs have arrived. A server controlled by a sequential policy waits for a random period upon becoming idle and starts sequentially transiting the sleep states in a decreasing order of power consumption until it reaches the deepest sleep state, where it waits for a threshold number of jobs to arrive before setup. We show for Poisson arrivals, generally distributed service times, setup delays and idle times that the optimal control of the idle time and sleep states neither randomizes nor transits sleep states sequentially. Either the system only uses the idle state and no sleep states, or the server immediately goes to some fixed sleep state and waits until a fixed number of jobs have arrived before setup. We prove this result for two cost metrics, namely weighted sum of energy and response time (ERWS) and their product (ERP).