Elastic Job Scheduling with Unknown Utility Functions

We consider a bipartite network consisting of job schedulers and parallel servers. Jobs arrive at the schedulers following stochastic processes with unknown arrival rates, and get routed to the servers, which execute the jobs with unknown service rates. The jobs are elastic, as their “size”, i.e., the amount of service needed for their completion, is determined by the schedulers. After a job finishes execution, some utility is obtained where the utility value depends on the job’s size through some underlying concave utility function. We consider the setting where the utility functions are unknown apriori, while a noisy observation of the utility value of each job is obtained upon its completion. Our goal is to design a policy that makes job-size and routing decisions to maximize the total utility obtained by the end of the time horizon T . We measure the performance of a policy by regret, i.e., the gap between the expected utility obtained under the policy and that under the optimal policy. We first establish an upper bound on the regret of a generic policy, that consists of the cumulative difference in utility between the jobsize decisions of the policy and the solution to a static optimization problem, and the total backlog of unfinished jobs at the end of the time horizon.We then propose a policy that simultaneously controls the cumulative utility difference and backlog of unfinished jobs, and achieves an order optimal regret of ?̃? ( √ T ). Our policy solves the elastic job scheduling problem by extending the Stochastic Convex Bandit Algorithm to handle unknown and stochastic constraints, and making routing decisions based on the Join-the-Shortest-Queue rule. It also presents a principled approach to extending algorithms for zeroth-order convex optimization to the settings with unknown and stochastic constraints.