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Queues with Rechargeable Servers

Eliezer Fuentes-Quezada, Jamol Pender

TL;DR

The paper introduces the Erlang--S^* queue to model service systems where servers intermittently become unavailable due to post-service charging, parameterized by a charging probability $p$ and return rate $\gamma$, with customer abandonment at rate $\theta$. It develops fluid limits that reveal a systematic capacity loss and diffusion limits that yield a two-dimensional Ornstein–Uhlenbeck process with a nontrivial covariance between queue length and active servers. These limits enable closed-form or easily computable staffing rules to meet delay and abandonment targets, including both deterministic and joint-normal (with covariance) approaches, and are validated by extensive simulations. The results provide actionable insights for energy-aware staffing in drone delivery and other systems with endogenous server unavailability, demonstrating substantial improvements over traditional fixed-capacity staffing. The framework also opens paths to extensions with heterogeneous fleets, state-dependent charging policies, and risk-sensitive performance objectives.

Abstract

Drone delivery systems violate a core assumption in classical queueing models: server capacity is not fixed. Drones (servers) periodically must recharge, creating random fluctuations in service availability. We introduce an Erlang--S$^{*}$ queue that incorporates charging dynamics (probability of charging after service completion $p$ and charging return rate $γ$) together with abandonment. We derive fluid and diffusion limits, yielding closed-form steady-state means, variances, and covariances for the joint queue--server process $(Q,S)$. The diffusion limits allow us to derive new staffing rules for the probability of delay and the probability of abandonment targets. A key insight is that server stochasticity induces systematic capacity loss relative to fixed--server systems, leading to a regime--dependent staffing adjustment: additive shifts in underloaded regimes and multiplicative scaling in overloaded regimes. Our simulation experiments confirm both the accuracy of the limit theorems and the performance of the staffing schedule's ability to achieve their targets.

Queues with Rechargeable Servers

TL;DR

The paper introduces the Erlang--S^* queue to model service systems where servers intermittently become unavailable due to post-service charging, parameterized by a charging probability and return rate , with customer abandonment at rate . It develops fluid limits that reveal a systematic capacity loss and diffusion limits that yield a two-dimensional Ornstein–Uhlenbeck process with a nontrivial covariance between queue length and active servers. These limits enable closed-form or easily computable staffing rules to meet delay and abandonment targets, including both deterministic and joint-normal (with covariance) approaches, and are validated by extensive simulations. The results provide actionable insights for energy-aware staffing in drone delivery and other systems with endogenous server unavailability, demonstrating substantial improvements over traditional fixed-capacity staffing. The framework also opens paths to extensions with heterogeneous fleets, state-dependent charging policies, and risk-sensitive performance objectives.

Abstract

Drone delivery systems violate a core assumption in classical queueing models: server capacity is not fixed. Drones (servers) periodically must recharge, creating random fluctuations in service availability. We introduce an Erlang--S queue that incorporates charging dynamics (probability of charging after service completion and charging return rate ) together with abandonment. We derive fluid and diffusion limits, yielding closed-form steady-state means, variances, and covariances for the joint queue--server process . The diffusion limits allow us to derive new staffing rules for the probability of delay and the probability of abandonment targets. A key insight is that server stochasticity induces systematic capacity loss relative to fixed--server systems, leading to a regime--dependent staffing adjustment: additive shifts in underloaded regimes and multiplicative scaling in overloaded regimes. Our simulation experiments confirm both the accuracy of the limit theorems and the performance of the staffing schedule's ability to achieve their targets.
Paper Structure (52 sections, 16 theorems, 126 equations, 21 figures, 3 tables)

This paper contains 52 sections, 16 theorems, 126 equations, 21 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions of the Paper
  3. Organization of the Paper
  4. The Erlang--S^* Model
  5. Stochastic Model Construction
  6. Stochastic Dynamics
  7. Explanation
  8. Fluid Approximation
  9. Steady-State Regimes
  10. Diffusion Limits and the Covariance Structure
  11. Underloaded Regime ($q^*<s^*$)
  12. Overloaded Regime ($q^*>s^*$)
  13. Example 1:
  14. Example 2:
  15. Staffing for Delay Targets
  16. ...and 37 more sections

Key Result

Theorem 3.1

Consider the scaled stochastic processes Then for every fixed $T>0$, where $(\bar{q},\bar{s})$ is the unique solution on $[0,T]$ of

Figures (21)

  • Figure 1: Schematic representation of the Erlang-$S^*$ model.
  • Figure : (a) $Q(t)$ and $S(t)$, $\lambda = 100,\, \mu = 5 , \, \theta = 1,\, p=0.1,\, \gamma = 0.5$ and $c=100$ (UL)
  • Figure : (a) Distribution of $Q$ and $S$ for UL parameters of Fig. \ref{['fig:QS-trajectories']}(a).
  • Figure : (a) Variance and covariance with $\lambda = 100,\, \mu = 5 , \, \theta = 1,\, p=0.1,\, \gamma = 0.5$ and $c=100$ (UL)
  • Figure : (a) $Q(t)$ and $S(t)$, $\lambda = 100,\, \mu = 5 , \, \theta = 1,\, p=0.1,\, \gamma = 0.5$ and $c=100$ (UL)
  • ...and 16 more figures

Theorems & Definitions (42)

  • Theorem 3.1: Fluid Limit for the Erlang--S Queue
  • proof
  • Theorem 3.2
  • proof
  • Remark
  • Theorem 4.1: Diffusion limit for the Erlang--S$^\ast$ queue
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 32 more