Queues with Rechargeable Servers

TL;DR

This paper models queueing systems with drones as servers that periodically recharge, deriving limits and staffing rules to manage capacity fluctuations and meet delay and abandonment targets.

Contribution

It introduces an Erlang--S* queue model incorporating drone recharge dynamics and provides analytical limits and staffing strategies for such systems.

Findings

Server stochasticity causes systematic capacity loss.

Derived staffing rules improve target achievement.

Simulation confirms theoretical accuracy and staffing effectiveness.

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 $\gamma$) 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.