[Paper Review] Efficiency-Reward Trade-Off in Queues with Dynamic Arrivals

Tianze Qu, Sushil Mahavir Varma|arXiv (Cornell University)|Jan 30, 2026

Advanced Queuing Theory Analysis0 citations

TL;DR

The paper analyzes a single-server queue with queue-length dependent arrivals and a general reward function, establishing a Pareto frontier between long-run reward and queue length. It shows how market size and the reward function’s curvature at capacity determine whether fully dynamic control is needed for optimal efficiency-reward trade-offs.

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.

Motivation & Objective

• Motivate dynamic arrival control in queues with endogenous arrivals driven by online platforms.
• Define a unified framework to trade off long-run reward against queueing efficiency.
• Characterize how market size and reward curvature affect optimal control structure and efficiency scaling.
• Provide universal lower bounds and policy constructions that achieve order-optimal queue-length scaling under regret constraints.

Proposed method

• Model the system as an M/M/1 queue with state-dependent arrival rate lambda(q) in [0, lambda_max] and fixed service rate mu=1.
• Define reward r(lambda) as the long-run average E[F(lambda(bar{q}))] and measure regret against the fluid benchmark F*.
• Use a two-step analysis: first derive a fluid benchmark solving a moment-constrained optimization, then design near-optimal state-dependent policies to match the benchmark.
• Prove universal lower bounds on E[bar{q}] and construct policies achieving these bounds in different regimes of lambda_max and F.
• Differentiate regimes based on whether F is concave-like at capacity (1) via a curvature condition, guiding policy design (fully dynamic vs. simpler policies).
• Relate results to classical heavy-traffic theory and prior dynamic pricing/revenue management literature.

Experimental results

Research questions

• RQ1 What is the fundamental trade-off between long-run reward and queue efficiency for a single-server queue with dynamic arrivals?
• RQ2 How does market size (lambda_max) affect the optimality gap between static, two-rate, and fully dynamic arrival policies?
• RQ3 How does the curvature of the reward function F around capacity influence the optimal control structure and queue-length scaling?
• RQ4 What are universal lower bounds on the steady-state queue length under an epsilon-regret constraint, and can we design policies that achieve them?
• RQ5 How do the results connect to and extend classical heavy-traffic and dynamic pricing literature?

Key findings

• If lambda_max <= 1 (small market), any admissible policy yields E[bar{q}] = Ω(1/ε), showing no efficiency gain from dynamic control.
• If lambda_max > 1 (large market) and F is concave-like around capacity, the optimal queue length scales as E[bar{q}] = Θ(1/√ε] with fully dynamic policies achieving the bound.
• If lambda_max > 1 and F is not concave-like around capacity, E[bar{q}] scales as Θ(log(1/ε)) with two-rate policies achieving this bound.
• Universal lower bounds show E[bar{q}] ≥ Ω(log(1/ε)) in the non-concave-like case and E[bar{q}] ≥ Ω(1/√ε) in the concave-like case under epsilon-regret constraints.
• Fully dynamic arrival control is necessary to achieve order-optimal efficiency for concave-like F, whereas simpler policies (static or two-rate) can be order-optimal in the non-concave-like regime.
• The work extends heavy-traffic theory to queues with dynamic arrivals and unifies pricing/revenue and throughput optimization under a single framework.