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.

翻译：本文研究多维情况下的云服务器租赁问题（RSiC）。在该问题中，一系列多参数任务需要在可按需租赁的服务器上进行调度。每个任务具有到达时间、完成时间以及一个多维规模向量（用于定义其资源需求）。每台服务器具有多维容量，只要在每个维度上任务规模之和不超过服务器在该维度的容量，任务即可被调度至该服务器。目标是最小化处理任务序列所需的服务器总租赁时间。AF算法在无需租赁新服务器处理任务时不会新增服务器。我们引入AF算法子类——单调AF算法，并证明该子类具有紧竞争比$\Theta(d \mu)$，其中$d$为问题维度，$\mu$为输入序列中任务最大与最小持续时间的比值。此外，我们证明RSiC问题的上界关于维度$d$满足直接和性质，即展示了如何将一维RSiC算法转化为可在$d$维环境下工作，且竞争比按因子$d$缩放。据此推论，我们获得$d$维可预见型RSiC的$O(d\sqrt{\log \mu})$上界。在$\mu \le \log d - 2$假设下，我们还建立了$d$维非可预见型RSiC的确定性及随机化算法的下界$\widetilde{\Omega}(d \mu)$。最后，我们提出名为Greedy的自然贪心算法——属于单调AF家族的可预见型算法，其竞争比为$\Theta(d \mu)$。实验结果表明，在我们实现的几乎所有到达率、$\mu$与$d$参数设置下，Greedy算法性能优于或匹配所有现有算法。