## Abstract

Electricity cost is a dominant and rapidly growing expense in data centers. Unfortunately, much of the consumed energy is wasted, because servers are idle for extended periods of time. We study a capacity management problem that dynamically right-sizes a data center, matching the number of active servers with the varying demand for computing capacity. We resort to a data-center optimization problem introduced by Lin, Wierman, Andrew, and Thereska [25, 27] that, over a time horizon, minimizes a combined objective function consisting of operating cost, modeled by a sequence of convex functions, and server switching cost. All prior work addresses a continuous setting in which the number of active servers, at any time, may take a fractional value. In this article, we investigate for the first time the discrete data-center optimization problem where the number of active servers, at any time, must be integer valued. Thereby, we seek truly feasible solutions. First, we show that the offline problem can be solved in polynomial time. Our algorithm relies on a new, yet intuitive graph theoretic model of the optimization problem and performs binary search in a layered graph. Second, we study the online problem and extend the algorithm Lazy Capacity Provisioning (LCP) by Lin et al. [25, 27] to the discrete setting. We prove that LCP is 3-competitive. Moreover, we show that no deterministic online algorithm can achieve a competitive ratio smaller than 3. Hence, while LCP does not attain an optimal competitiveness in the continuous setting, it does so in the discrete problem examined here. We prove that the lower bound of 3 also holds in a problem variant with more restricted operating cost functions, introduced by Lin et al. [25]. In addition, we develop a randomized online algorithm that is 2-competitive against an oblivious adversary. It is based on the algorithm of Bansal et al. [7] (a deterministic, 2-competitive algorithm for the continuous setting) and uses randomized rounding to obtain an integral solution. Moreover, we prove that 2 is a lower bound for the competitive ratio of randomized online algorithms, so our algorithm is optimal. We prove that the lower bound still holds for the more restricted model. Finally, we address the continuous setting and give a lower bound of 2 on the best competitiveness of online algorithms. This matches an upper bound by Bansal et al. [7]. A lower bound of 2 was also shown by Antoniadis and Schewior [4]. We develop an independent proof that extends to the scenario with more restricted operating cost.

Original language | English |
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Article number | 15 |

Journal | ACM Transactions on Parallel Computing |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - 16 Dec 2022 |

## Keywords

- Homogeneous servers
- discrete setting
- lower bounds
- online algorithm
- polynomial-time offline algorithm