Data centres and similar server clusters consume a large amount of energy. However, not all consumed energy produces useful work. Servers consume a disproportional amount of energy when they are idle, underutilised, or overloaded. The effect of these conditions can be minimised by attempting to balance the demand for and the supply of resources through a careful prediction of future workloads and their efficient consolidation. In this paper we extend the cutting stock problem for consolidating workloads having stochastic characteristics. Hence, we employ the aggregate probability density function of co-located and simultaneously executing services to establish valid patterns. A valid pattern is one yielding an overall resource utilisation below a set threshold. We tested the scope and usefulness of our approach on a 16-core server with 29 different benchmarks. The workloads of these benchmarks have been generated based on the CPU utilisation traces of 100 real-world virtual machines which we obtained from a Google data centre hosting more than 32000 virtual machines. Altogether, we considered 600 different consolidation scenarios during our experiment. We compared the performance of our approach-system overload probability, job completion time, and energy consumption-with four existing/proposed scheduling strategies. In each category, our approach incurred a modest penalty with respect to the best performing approach in that category, but overall resulted in a remarkable performance clearly demonstrating its capacity to achieve the best trade-off between resource consumption and performance.

We present a new approach for analog receive beamforming if phase shifters and amplifiers have finite resolution only. Then, the maximization of the signal-to-interference-plus-noise ratio (SINR) is a discrete optimization problem with a nonconcave objective function. The discrete maximization problem is solved exactly by means of a branch-and-bound algorithm. Based on the Capon method, we derive a new and efficient way of computing upper SINR-bounds for the subproblems occurring at the nodes of the branch-and-bound tree. Results of numerical simulations are provided and compared to an earlier approximate approach.