Renting Servers in the Cloud: Parameterized Analysis of FirstFit.

We study the renting servers in the cloud problem (RSiC), which is motivated by job allocation to servers in cloud computing applications. Jobs arrive in an online manner and the size of a job as well as its duration is known at the time of its arrival. All jobs must be assigned to servers, which can be rented on demand and each server has a limited capacity per unit of time. The number of available servers is unlimited, and the goal is to minimize the sum of rental times of servers. A natural algorithm for this problem is FirstFit, which greedily assigns a new job to the first server which is active and can accommodate the job at the time of its arrival (if no such server exists, a new server is rented). In spite of being conceptually simple, FirstFit is notoriously difficult to analyze for many packing problems, indicating a lack of suitable techniques for such an analysis. In this paper, we approach analysis of FirstFit for RSiC from a parameterized perspective: we consider families of inputs which result in FirstFit using at most k servers at a time (for various values of k) and establish tight bounds on the competitiveness of FirstFit on such inputs. The parameterized version of the problem has a natural interpretation, namely, it describes the scenario when the number of available servers is limited but sufficient to handle the incoming demand. We establish a tight competitive ratio of 2 when k = 2 and a tight competitive ratio of 3 when k = 3 or k = 4. In particular, our results improve the previous lower bound of 2.518 to 3 for the case of RSiC of equal duration jobs in the general setting, narrowing the gap between the best known upper bound of 4 and the lower bound to just 1. We also performed a thorough experimental study of FirstFit from the same parameterized perspective with respect to random inputs.