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

Abstract

Abstract. We consider Markovian many-server systems with admission control operating in a Quality-and-Efficiency-Driven (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.

Optimality gaps in asymptotic dimensioning of many-server systems

“…Dividing by the factor (sρ) s /s!, we obtain the form In [4,7], it is proven using Jagerman's asymptotic expansions [3] for Erlang B's formula that asymptotically,…”

“…Ref. [7] discusses the problem of maximizing the revenue rate over the set of all admission control policies. One advantage of considering general admission control policies and revenue structures is that one can study different service systems and steadystate performance measures through one unifying framework.…”

“…We now introduce joint dimensioning and admission control, which has not been studied in the QED literature. In [7], it is proven that the admission control policy that maximizes the system's revenue rate has a threshold structure, and that for fixed γ < ∞ there exists an optimal threshold level η opt . In §4.1, we fixed η < ∞ and searched for γ opt .…”

“…For a wide range of performance metrics, M s (γ) say, these refinements are of the form M s (γ) = M 0 (γ) + M 1 (γ)/ √ s + · · · . The method in [4,7] can deliver as many higher-order terms as needed, and generate all the refinements obtained in [5,8,6].…”

“…In the present paper, we demonstrate how the results in [4,7] can be leveraged to determine the optimality gaps of novel asymptotic dimensioning problems for a large class of many-server systems. Our main result (Theorem 1) provides generic bounds for the optimality gaps that become sharper when more terms in the QED expansion for M s (γ) are included.…”