Renting Servers for Multi-Parameter Jobs in the Cloud
Yaqiao Li, Mahtab Masoori, Lata Narayanan, Denis Pankratov
TL;DR
The paper studies online, multi-parameter renting of cloud servers (RSiC) in $d$ dimensions, focusing on how to minimize total server rental time for a sequence of jobs with arrivals, durations, and multi-dimensional resource demands. It introduces monotone AnyFit algorithms and a clairvoyant Greedy variant, proving an upper bound of $\rho \le 3\mu d+1$ for this family and establishing a direct-sum property that lifts any 1D RSiC algorithm to $d$ dimensions with a factor of $d$, yielding a clairvoyant bound of $O(d\sqrt{\log \mu})$ in the multi-dimensional setting. The work also derives lower bounds via online graph coloring for both deterministic and randomized algorithms in non-clairvoyant and clairvoyant models, showing $\tilde{\Omega}(d\mu)$ in several regimes. An extensive experimental study on synthetic data demonstrates that Greedy often matches or outperforms existing approaches across varied arrival rates, durations, and dimensionalities. Together, these results advance understanding of online, multi-parameter cloud resource allocation, offering practical strategies and fundamental limits for both clairvoyant and non-clairvoyant settings.
Abstract
We study the Renting Servers in the Cloud problem (RSiC) in multiple dimensions. In this problem, a sequence of multi-parameter jobs must be scheduled on servers that can be rented on-demand. Each job has an arrival time, a finishing time, and a multi-dimensional size vector that specifies its resource demands. Each server has a multi-dimensional capacity and jobs can be scheduled on a server as long as in each dimension the sum of sizes of jobs does not exceed the capacity of the server in that dimension. The goal is to minimize the total rental time of servers needed to process the job sequence. AF algorithms do not rent new servers to accommodate a job unless they have to. We introduce a sub-family of AF algorithms called monotone AF algorithms. We show this family have a tight competitive ratio of $Theta(d mu)$, where $d$ is the dimension of the problem and $mu$ is the ratio between the maximum and minimum duration of jobs in the input sequence. We also show that upper bounds for the RSiC problem obey the direct-sum property with respect to dimension $d$, that is we show how to transform $1$-dimensional algorithms for RSiC to work in the $d$-dimensional setting with competitive ratio scaling by a factor of $d$. As a corollary, we obtain an $O(d\sqrt{log mu})$ upper bound for $d$-dimensional clairvoyant RSiC. We also establish a lower bound of $\widetilde{Omega}(d mu)$ for both deterministic and randomized algorithms for $d$-dimensional non-clairvoyant RSiC, under the assumption that $mu \le log d - 2$. Lastly, we propose a natural greedy algorithm called Greedy. Greedy, is a clairvoyant algorithm belongs to the monotone AF family, achieves a competitive ratio of $Theta(d mu)$. Our experimental results indicate that Greedy performs better or matches all other existing algorithms, for almost all the settings of arrival rates and values of mu and $d$ that we implemented.