Asymptotic Optimality of $c\mu$-Rule for LLM Inference Scheduling under Memory Constraint

Abstract

Large language model (LLM) inference, the process by which a pre-trained model receives an input prompt and generates output tokens sequentially, has become increasingly important with the widespread deployment of LLM-based applications. As inference workloads continue to scale, GPU memory has emerged as a critical bottleneck: the long input prompts and generation sequences common in agentic and reasoning workloads lead to substantial key-value (KV) cache usage. In this paper, we study the resulting online scheduling problem from a queueing perspective. Requests arrive stochastically with heterogeneous prompt and output lengths, undergo prefill and autoregressive decode, and compete for a finite KV-cache capacity. Unlike classical queueing models, the duration of a processing step depends on both prefill computation and KV-cache I/O, so service capacity is jointly determined by computation, memory access, and memory occupancy. We develop a fluid model that reduces the system to a multiclass single-server queue with effective service rates capturing computation, memory access, and memory capacity. This fluid analysis yields a $c\mu$ priority rule, where the effective service rate $\mu$ is jointly determined by computational speed, memory bandwidth, and memory capacity. We prove that this rule minimizes cumulative holding cost in the fluid model and, for a finite workload, also minimizes completion time and maximizes full-run output-token throughput. We then construct a feasible scheduling policy for the original finite system and prove that its scaled dynamics converge to the optimal fluid policy. Finally, experiments on a real LLM serving engine using an NVIDIA H800 show that practical implementations of the proposed priority rule substantially reduce average latency while maintaining high throughput relative to existing scheduling policies.