An $Ω(\log(N)/N)$ Lookahead is Sufficient to Bound Costs in the Overloaded Loss Network
Robert L. Bray
TL;DR
This work analyzes admission control in an overloaded loss network with two customer classes and reusable resources, framing the online regret relative to fluid and offline benchmarks. It decomposes costs into volatility and uncertainty components, proving the volatility cost is bounded while the uncertainty cost scales as Θ(log N); crucially, an Ω(log N / N) lookahead is enough to bound both long-run and short-run uncertainty costs, effectively aligning offline and fluid benchmarks as N grows. The study introduces the Promise of Future Idleness (PFI) to bound volatility via a renewal-reward analysis and proposes a Single Short Sensor (SSS) lookahead policy that achieves a comparable bound for long-run uncertainty, demonstrating asymptotic equivalence of benchmarks under finite forecasting. Together, these results provide a precise, practically relevant threshold for the amount of future information needed to control costs in revenue management with reusable resources.
Abstract
I study the simplest model of revenue management with reusable resources: admission control of two customer classes into a loss queue. This model's long-run average collected reward has two natural upper bounds: the deterministic relaxation and the full-information offline problem. With these bounds, we can decompose the costs faced by the online decision maker into (i) the \emph{cost of variability}, given by the difference between the deterministic value and the offline value, and (ii) the \emph{cost of uncertainty}, given by the difference between the offline value and the online value. \cite{Xie2025} established that the sum of these two costs is $Θ(\log N)$, as the number of servers, $N$, goes to infinity. I show that we can entirely attribute this $Θ(\log N)$ rate to the cost of uncertainty, as the cost of variability remains $O(1)$ as $N \rightarrow \infty$. In other words, I show that anticipating future fluctuations is sufficient to bound operating costs -- smoothing out these fluctuations is unnecessary. In fact, I show that an $Ω(\log(N)/N)$ lookahead window is sufficient to bound operating costs.