Server farms today consume more than 1.5% of the total electricity in the U.S. at a cost of nearly $4.5 billion. Given the rising cost of energy, many industries are now seeking solutions for how to best make use of their available power. An important question which arises in this context is how to distribute available power among servers in a server farm so as to get maximum performance. By giving more power to a server, one can get higher server frequency (speed). Hence it is commonly believed that, for a given power budget, performance can be maximized by operating servers at their highest power levels. However, it is also conceivable that one might prefer to run servers at their lowest power levels, which allows more servers to be turned on for a given power budget. To fully understand the effect of power allocation on performance in a server farm with a fixed power budget, we introduce a queueing theoretic model, which allows us to predict the optimal power allocation in a variety of scenarios. Results are verified via extensive experiments on an IBM BladeCenter. We find that the optimal power allocation varies for different scenarios. In particular, it is not always optimal to run servers at their maximum power levels. There are scenarios where it might be optimal to run servers at their lowest power levels or at some intermediate power levels. Our analysis shows that the optimal power allocation is non-obvious and depends on many factors such as the power-to-frequency relationship in the processors, the arrival rate of jobs, the maximum server frequency, the lowest attainable server frequency and the server farm configuration. Furthermore, our theoretical model allows us to explore more general settings than we can implement, including arbitrarily large server farms and different power-to-frequency curves. Importantly, we show that the optimal power allocation can significantly improve server farm performance, by a factor of typically 1.4 and as much as a factor of 5 in some cases.

The M/M/k/setup model, where there is a penalty for turning servers on, is common in data centers, call centers and manufacturing systems. Setup costs take the form of a time delay, and sometimes there is additionally a power penalty, as in the case of data centers. While the M/M/1/setup was exactly analyzed in 1964, no exact analysis exists to date for the M/M/k/setup with k > 1.In this paper we provide the first exact, closed-form analysis for the M/M/k/setup and some of its important variants including systems in which idle servers delay for a period of time before turning off or can be put to sleep. Our analysis is made possible by our development of a new technique, Recursive Renewal Reward (RRR), for solving Markov chains with a repeating structure. RRR uses ideas from renewal reward theory and busy period analysis to obtain closed-form expressions for metrics of interest such as the transform of time in system and the transform of power consumed by the system. The simplicity, intuitiveness, and versatility of RRR makes it useful for analyzing Markov chains far beyond the M/M/k/setup. In general, RRR should be used to reduce the analysis of any 2-dimensional Markov chain which is infinite in at most one dimension and repeating to the problem of solving a system of polynomial equations. In the case where all transitions in the repeating portion of the Markov chain are skip-free and all up/down arrows are unidirectional, the resulting system of equations will yield a closed-form solution.