Feedback Control for Adaptive Language Model Routing Under Non-Stationary Workloads
Serving a stream of requests across large language models (LLMs) of differing cost and quality is an online allocation problem, usually framed as multi-armed bandits. We frame it as feedback control: a direct-acting Proportional–Integral–Derivative (PID) controller whose setpoint is the running fleet- average performance and whose bounded output adjusts each model’s allocation share, with requests routed by weighted sampling over the allocation vector. This work contributes (i) a stability analysis ofthe closed loop — bounded-input bounded-output behaviour by anti-windup, exponential convergence of the performance estimates via a Lyapunov function, and a persistent-excitation condition guaranteeing recoverability after a regime change; (ii) a closed-form, analysis-grounded automatic tuning rule requiring no per-dataset search; and (iii) an honest head-to-head against static, round-robin, random, epsilon-greedy, UCB1, Thompson sampling, and the non-stationary bandits Sliding-Window UCB and Discounted UCB, on GSM8K with a checkable exact-match reward, reporting inference cost and request latency alongside quality. Under transient drift the auto-tuned controller is statistically tied with the best non-stationary bandit at lower cost; under a persistent regime shift it significantly outperforms both (p < 0.03). We further show the integral term helps only under a persistent shift — a proportional controller suffices for transient drift — and evaluate robustness to noisy rewards. The result is an interpretable, stability-analyzed, auto-tuned routing controller competitive with or superior to non- stationary bandits at lower cost.