Structural properties and exact analysis of energy-aware multiserver queueing systems with setup times

Energy consumption of today's datacenters is a constant concern from the standpoints of monetary and environmental costs. An intuitive solution to address these immense energy demands is to turn servers off to incur less costs. As such, many different authors have modelled this problem as an M∕M∕C queue where each server can be turned on, with an exponentially distributed setup time, or turned off instantaneously. What policy the model should employ, or rather when each server should be turned on and off is far from a trivial question. A specific policy is often examined, but determining which policy to study can be a difficult process and is often a product of intuition. Moreover, while a specific policy may do comparatively well against another, in general it may be far from optimal. This problem is further accentuated when one considers the case that a policy may do well or even be optimal under a specific cost function, but far from optimal under another. To address this issue we study the structural properties of the optimal policy under linear cost functions, allowing for a significant reduction in the search space. We then leverage these structural properties to intelligently select two policies for further study. Using the recursive renewal reward technique, we offer an exact analysis of these policies alongside offering insights, observations, and implications for how these systems behave. In particular, we provide insight into the question of the number of servers that should remain on at all times under a general cost function.