Modern machine learning inference systems often host multiple models that can perform the same task with different levels of accuracy and latency. For example, a large model can be more accurate but slow, whereas a smaller and less accurate can be faster in serving inference queries. Amidst the rapid advancements in Large Language Models (LLMs), it is paramount for such systems to strike the best trade-off between latency and accuracy. In this paper, we consider the problem of designing job assignment policies for a multi-server queueing system where servers have heterogeneous rates and accuracies, and our goal is to minimize the expected inference latency while meeting an average accuracy target. Such queueing systems with constraints have been sparsely studied in prior literature to the best of our knowledge. We first identify a lower bound on the minimum achievable latency under any policy that achieves the target accuracy using a linear programming (LP) formulation. Building on the LP solution, we introduce a Randomized-Join-the Idle Queue (R-JIQ) policy, which consistently meets the accuracy target and asymptotically (as system size increases) achieves the optimal latency . However, the R-JIQ policy relies on the knowledge of the arrival rate to solve the LP. To address this limitation, we propose the Prioritize Ordered Pairs (POP) policy that incorporates the concept of ordered pairs of servers into waterfilling to iteratively solve the LP. This allows the POP policy to function without relying on the arrival rate. Experiments suggest that POP performs robustly across different system sizes and load scenarios, achieving near-optimal performance.
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