Optimal Parallel Scheduling under Concave Speedup Functions

Efficient scheduling of parallel computation resources across multiple jobs is a fundamental problem in modern cloud/edge computing systems for many AI-based applications. Allocating more resources to a job accelerates its completion, but with diminishing returns. Prior work (heSRPT) solved this problem only for some specific speedup functions with an exponential form, providing a closed-form solution. However, the general case with arbitrary concave speedup functions -- which more accurately capture real-world workloads -- has remained open. In this paper, we solve this open problem by developing optimal scheduling algorithms for parallel jobs under general concave speedup functions. We first discover a fundamental and broadly-applicable rule for optimal parallel scheduling, namely the Consistent Derivative Ratio (CDR) Rule, which states that the ratio of the derivatives of the speedup functions across active jobs remains constant over time. To efficiently compute the optimal allocations that satisfy the CDR Rule, we propose the General Water-Filling (GWF) method, a more general version of classical water-filling in wireless communications. Combining these insights, we design the SmartFill Algorithm to solve the general scheduling problem. Unlike heSRPT, which always allocates resources to all active jobs, SmartFill selectively determines which jobs should receive resources and how much they should be allocated. For a broad class of so-called \emph{regular} speedup functions, SmartFill yields closed-form optimal solutions, while for non-regular functions it efficiently computes the optimum with low complexity. Numerical evaluations show that SmartFill can substantially outperform heSRPT across a wide range of concave speedup functions.