Optimal Admission Control for Many-Server Systems with QED-Driven Revenues

TL;DR

This paper analyzes optimal admission control in large many-server systems operating in a QED regime, identifying threshold policies that maximize revenue as system size grows, with explicit solutions for certain revenue functions.

Contribution

It introduces a limit optimization framework for revenue maximization in QED regimes and characterizes optimal admission thresholds, including explicit solutions for specific revenue structures.

Findings

Optimal thresholds scale with the square-root of system size.

Threshold policies outperform simple admission rules.

Explicit formulas derived for linear and exponential revenue models.

Abstract

We consider Markovian many-server systems with admission control operating in a QED regime, where the relative utilization approaches unity while the number of servers grows large, providing natural Economies-of-Scale. In order to determine the optimal admission control policy, we adopt a revenue maximization framework, and suppose that the revenue rate attains a maximum when no customers are waiting and no servers are idling. When the revenue function scales properly with the system size, we show that a nondegenerate optimization problem arises in the limit. Detailed analysis demonstrates that the revenue is maximized by nontrivial policies that bar customers from entering when the queue length exceeds a certain threshold of the order of the typical square-root level variation in the system occupancy. We identify a fundamental equation characterizing the optimal threshold, which we extensively leverage to provide broadly applicable upper/lower bounds for the optimal threshold, establish its monotonicity, and examine its asymptotic behavior, all for general revenue structures. For linear and exponential revenue structures, we present explicit expressions for the optimal threshold.