Fluid Limits for Time-Varying Many-Server Queues with Finite Capacity

TL;DR

This paper derives fluid limit models for nonstationary many-server queues with finite capacity, providing a rigorous foundation for understanding their transient behavior and guiding optimal staffing and capacity decisions.

Contribution

It introduces a novel fluid limit framework for time-varying many-server loss systems with general service times, including finite-buffer regimes, with well-posed nonlinear integral equations.

Findings

Fluid limits characterized by nonlinear Volterra equations.

Convergence of acceptance probabilities in transient regimes.

Numerical simulations confirm theoretical convergence and insights.

Abstract

This paper develops fluid limits for nonstationary many-server loss systems with general service-time distributions. For the zero-buffer $M_t/G/n/n$ queuing model, we prove a functional strong law of large numbers for the fraction of busy servers and characterize the limit by a nonlinear Volterra integral equation with discontinuous coefficients induced by instantaneous blocking. Well-posedness is established through an appropriate solution concept, yielding the time-varying acceptance probability without heuristic approximations. We then treat the finite-buffer $M_t/G/n/(n+b_n)$ regime, proving a functional strong law of large numbers for the triplet of fractions of busy servers, occupied buffers, and cumulative departures, whose limit satisfies a coupled system of three discontinuous Volterra equations capturing the interaction of service completions, buffer occupancy, and admission control at the capacity boundary. We establish well-posedness and convergence of the time-varying acceptance probability. Our theoretical results are supported by numerical simulations for both zero and finite-buffer regimes, illustrating the convergence of transient acceptance probabilities guaranteed by our theory. Finally, we use the fluid limits to derive optimal staffing and buffer-capacity for both time-varying loss systems.