On Delay-Optimal Scheduling in Queueing Systems with Replications

In modern computer systems, long-running jobs are divided into a large number of short tasks and executed in parallel. Experience in practical systems suggests that task service times are highly random and the job service latency is bottlenecked by the slowest straggling task. One common solution for straggler mitigation is to replicate a task on multiple servers and wait for one replica of the task to finish early. The delay performance of replications depends heavily on the scheduling decisions of when to replicate, which servers to replicate on, and which task to serve first. So far, little is understood on how to optimize these scheduling decisions for minimizing the job service latency. In this paper, we present a comprehensive theoretical analysis on delay-optimal scheduling in queueing systems with replications. In particular, low-complexity replication policies are designed, and are rigorously proven to be delayoptimal or near delay-optimal among all non-preemptive and causal policies. These theoretical results are established for very general system settings and delay metrics which allow for arbitrary arrival process, arbitrary job sizes, arbitrary soft deadlines, and heterogeneous servers with data locality constraints. In order to prove these results, novel sufficient conditions are developed for sample-path delay optimality and near delay optimality, which can be applied to any queueing system and are not limited to the study of replications. Index Terms Queueing systems with replications, delay optimality, data locality, sample-path ordering, workefficiency ordering, stochastic ordering. Yin Sun and C. Emre Koksal are with the Department of Electrical and Computer Engineering, the Ohio State University, Columbus, OH. Email: [email protected], [email protected]. Ness B. Shroff is with the Departments of Electrical and Computer Engineering and Computer Science and Engineering, the Ohio State University, Columbus, OH. Email: [email protected].