Stochastic Load Balancing with Machine Reservations

2026-07-27

AI Summary

This work investigates the performance trade-offs of non-adaptive strategies in stochastic load balancing. It proposes a two-stage model: in the first stage, each job reserves up to $k$ machines based on the task size distribution; in the second stage, after observing the actual job size, it is assigned to one of the reserved machines to minimize the expected makespan. The paper establishes, for the first time in this setting, a “power of two choices” theory, showing that under identical machines, reserving just two machines per job suffices to achieve a constant-factor approximation to the omniscient optimal solution. For related machines, it provides an $O(\log m / \log \log m)$-approximation and a bicriteria constant-factor approximation, and further proves that with 2-reservation, one can approximate the adaptively optimal solution.

Abstract

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.

Problem

• stochastic load balancing
• machine reservations
• non-adaptive policies
• makespan minimization
• power-of-two-choices

Innovation

• stochastic load balancing
• machine reservations
• power of two choices
• non-adaptive policies
• makespan minimization