Motivated primarily by applications in cloud computing, we study a simple, yet powerful, online allocation problem in which jobs of varying durations arrive over continuous time and must be assigned immediately and irrevocably to one of the available offline servers. Each server has a fixed initial capacity, with assigned jobs occupying one unit for their duration and releasing it upon completion. The algorithm earns a reward for each assignment upon completion. We consider a general heterogeneous setting where both the reward and duration of a job depend on the job-server pair. The objective of the online algorithm is to maximize the total collected reward, and remain competitive against an omniscient benchmark that knows all job arrivals in advance. Our main contribution is the design of a new online algorithm, termed Forward-Looking BALANCE (FLB), and using primal-dual framework to establish that it is (asymptotically) optimal-competitive. This meta-algorithm has two main primitives: (i) keeping track of the capacity used for each server at each time and applying a penalty function to this quantity, and (ii) adjusting the reward of assigning a job to a server by subtracting the total penalty of a particularly chosen subset of future times, in contrast to just looking at the current time. The FLB algorithm then assigns the arriving job to the server with the maximum adjusted reward. If R and D are the ratios of maximum over minimum rewards and durations, we show that the FLB algorithm obtains an asymptotic competitive ratio of ln(RD)+3lnln(max(R,D))+O(1). We further show this bound has optimal dependencies on all the parameters. Our main analysis combines a novel dual-fitting technique, which leverages the configuration LP benchmark for this problem, and a novel inductive argument to establish the capacity feasibility of the algorithm, which might be of independent interest.

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