When Should Temporary Capacity Be Added? Optimal Timing of Short-Duration Staffing Boosts in Time-Varying Queues, and Why Fluid Approximations Overstate the Benefit

Service systems facing a predictable demand peak (a lunch rush, a call-center surge) often respond by temporarily adding capacity — a second cashier, an extra agent — for a fixed duration during the busy period. The standard heuristic is to center this temporary capacity on the demand peak. We show, both analytically via a deterministic fluid approximation and computationally via discrete-event simulation of the underlying stochastic M_t/M/c(t) queue, that this heuristic is measurably suboptimal in most of the parameter regimes we study: because a queue that has already built up takes time to drain, the total-customer-wait-minimizing window is typically shifted later than the demand peak, not centered on it. Across three demand-shape/budget-duration scenarios, a properly randomized, variance-reduced discrete-event simulation (600–1,500 replications with common random numbers) confirms this direction and finds it statistically significant in every case, while a matched deterministic fluid approximation overstates the magnitude of the benefit by a factor of roughly 7–12× in every scenario tested. This overstatement is consistent with, and gives a concretely quantified instance of, existing asymptotic theory showing that fluid and pointwise-stationary approximations for time-varying queues are only exact in the large-rate (many-server) limit; our moderate-scale, single-server setting is exactly where that theory predicts such approximations should be least reliable. We also identify a boundary regime (a narrow demand peak combined with a long capacity budget) in which the direction of the optimal shift reverses, underscoring that the heuristic's failure mode is itself regime-dependent.