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

Mingrui Wang, Prakash Chakraborty

TL;DR

The paper addresses nonstationary, non-Markovian loss queues with time-varying demand by developing fluid limits that are governed by nonlinear discontinuous Volterra integral equations. It analyzes both zero-buffer $M_t/G/n/n$ and finite-buffer $M_t/G/n/(n+b_n)$ regimes, establishing functional strong laws of large numbers and well-posed coupled limit systems that yield time-varying acceptance probabilities and allow direct performance computation without simulation. The results provide a rigorous basis for transient staffing and buffer-capacity design in applications such as call centers and emergency departments, and they connect to practical operational decisions through numerically tractable deterministic models. Overall, the framework unifies the analysis of time-dependent Erlang loss systems with general service-time distributions and offers a computational tool for real-time capacity planning under fluctuating demand.

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.

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

TL;DR

The paper addresses nonstationary, non-Markovian loss queues with time-varying demand by developing fluid limits that are governed by nonlinear discontinuous Volterra integral equations. It analyzes both zero-buffer and finite-buffer regimes, establishing functional strong laws of large numbers and well-posed coupled limit systems that yield time-varying acceptance probabilities and allow direct performance computation without simulation. The results provide a rigorous basis for transient staffing and buffer-capacity design in applications such as call centers and emergency departments, and they connect to practical operational decisions through numerically tractable deterministic models. Overall, the framework unifies the analysis of time-dependent Erlang loss systems with general service-time distributions and offers a computational tool for real-time capacity planning under fluctuating demand.

Abstract

This paper develops fluid limits for nonstationary many-server loss systems with general service-time distributions. For the zero-buffer 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 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.

Paper Structure

This paper contains 27 sections, 17 theorems, 232 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Overview of Approach and Key Insights.
  3. Contributions.
  4. Paper Organization.
  5. Preliminaries and Notations
  6. Convergence in Skorokhod Space.
  7. Counting Measure.
  8. Discontinuous Volterra Integral Equation.
  9. Notations.
  10. Fluid limit for zero-buffer loss system
  11. Setup.
  12. Fraction of Occupied Servers or Scaled Number in System.
  13. Fluid Limit for Loss System with Buffer
  14. Setup.
  15. Characterization of Relevant Stochastic Processes.
  16. ...and 12 more sections

Key Result

Theorem 2.1

A sequence of probability measures $\{\mathbb{P}_n\}$ on $(\mathbb{D},\mathcal{D})$ is tight if and only if: and for each $\varepsilon>0$,

Figures (9)

  • Figure 1: Zero-buffer loss system and its fluid model
  • Figure 2: Loss system with buffer and its fluid model
  • Figure 3: Zero-Buffer Loss Queue with Periodic Arrival Rate
  • Figure 4: Zero-Buffer Loss Queue with Episodic Arrival Rate
  • Figure 5: Finite Buffer Loss Queue with Periodic Arrival Rate
  • ...and 4 more figures

Theorems & Definitions (40)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.1
  • Theorem 2.3
  • Definition 2.2
  • Remark 2.1
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 30 more