Efficiency-Reward Trade-Off in Queues with Dynamic Arrivals

TL;DR

This work analyzes a single-server queue with state-dependent arrivals, formulating a regret-constrained optimization that trades off long-run reward $\\mathbb{E}[F(\\lambda(\\bar{q}))]$ against congestion $\\mathbb{E}[\\bar{q}]$ in an $M_q/M/1$ setting with $\\mu=1$. A fluid benchmark $F^{\\star}$ and an $\\varepsilon$-reward constraint yield a clear small-vs-large market dichotomy: in small markets ($\\lambda_{\\max}=1$) dynamic arrivals offer no efficiency gain, while in large markets ($\\lambda_{\\max}>1$) the optimal scaling depends on the curvature of $F$ at capacity. If $F$ is concave-like near capacity, the optimal queue length scales as $q^{\\star}=\\Theta(1/\\sqrt{\\varepsilon})$ and fully dynamic arrival control is necessary to achieve this limit; if $F$ is not concave-like, then $q^{\\star}=\\Theta(\\log(1/\\varepsilon))$ and a two-point policy suffices. The paper provides universal lower bounds, constructive fully dynamic policies, and a local-polyhedral dual interpretation, with applications to dynamic pricing in service systems and extensive numerical experiments validating the theory. The results offer non-asymptotic heavy-traffic insights and practical guidelines for designing efficient pricing and admission-control policies under general, possibly non-concave reward structures.

Abstract

Motivated by applications in online marketplaces such as ride-hailing platforms and payment channel networks, we study a single-server queue with state-dependent arrival control. The service operator dynamically chooses the arrival rate as a function of the current queue length and receives a reward determined by the induced rate, capturing objectives such as throughput, revenue, or social welfare. The goal is to design control policies that simultaneously achieve high long-run operating reward and low congestion, measured by the expected steady-state queue length. We adopt a regret-based framework relative to an optimal benchmark and characterize the efficiency--reward trade-off under an $\varepsilon$-optimal reward constraint. Our results reveal a sharp dichotomy between small-market and large-market regimes. In small markets, including state-independent policies, any admissible control incurs poor efficiency, with the expected queue length growing on the order of $1/\varepsilon$. In contrast, in large markets, state-dependent policies can achieve substantially better performance. When the reward function exhibits sufficient curvature, the optimal queue length scales as $Θ(1/\sqrt{\varepsilon})$; otherwise, it scales as $Θ(\log(1/\varepsilon))$. For each regime, we establish universal lower bounds on the achievable efficiency and construct simple state-dependent policies that attain these bounds. Our results provide a non-asymptotic heavy-traffic characterization for queues with dynamic arrivals and offer structural insights into the design of efficient pricing and admission control policies.

Figures (7)

• Figure 1: M/M/1 Birth--Death Chain with State-Dependent Arrivals
• Figure 2: Figure for two conditions
• Figure 3: Comparison of arrival policies.
• Figure 4: Construction of a quadratic function $G$
• Figure 5: Tradeoff comparison between two-arrival and fully dynamic policies

Theorems & Definitions (33)

• proposition 1
• proposition 2
• proposition 3
• theorem 1
• theorem 2
• theorem 3
• proposition 4
• proof : Proof of \ref{['prop:strictly_concave']}
• theorem 4
• proof