Stochastic Load Balancing with Machine Reservations

We introduce a novel variant of stochastic load balancing that enables a quantitative tradeoff between the practical benefits of non-adaptive policies and their performance limitations. Our model describes a solution in two stages. In the first stage, given only job-size distributions, we reserve a set of at most k machines for each job (a k-reservation). In the second stage, after observing job-size realizations, we assign each job to one of its reserved machines (a consistent assignment). The goal is to minimize the expected makespan. If k=1, we get the standard stochastic load balancing problem of finding a non-adaptive assignment with minimum expected makespan. If k is equal to the number of machines, then we obtain an all-powerful omniscient optimum that can tailor the assignment arbitrarily to the job-size realizations. We give a number of results that quantify this tradeoff. Most saliently, we show that in the setting of identical machines, a 2-reservation suffices to achieve a constant-factor approximation to the omniscient optimum, establishing a "power-of-two-choices" result for stochastic load balancing. We also show that this no longer holds true in the more challenging setting of related machines. Nonetheless, we give a number of positive algorithmic results for this setting: a true O(log m/log log m)-approximation; a bicriteria O(1)-approximation by reserving twice as many machines per job relative to an optimal k-reservation; and a 2-reservation whose cost is within a constant factor of what the adaptive optimum can achieve.