Staffing Many-Server Systems with Admission Control and Retrials

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 asymp-totic behavior, all for general revenue structures. For linear and exponential revenue structures, we present explicit expressions for the optimal threshold.

Scaled control in the QED regime

“…so that the stationary distribution exists if and only if (2.5) holds. The case lim ρ↑1/Ps F s (ρ) < ∞ (as considered for example in [FA95,JvL12]) can also be considered in the present context, but leads to some complications that distract attention from the bottom line of the exposition.…”

“…In §4, specifically Theorem 4.3, we show that (3.17) equals lim s→∞ D s (ρ), as expected, and we also derive the most relevant correction terms for finite s. Important here is that D s (ρ) converges to a value in the interval (0, 1) as s → ∞, which confirms that the local control in (3.5) leads to a nondegenerate limit. In [JvL12], s-independent control policies have been considered for which D s (ρ) has 1/ √ s-behavior for large s.…”