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<p>Motivated by the need to manage customers’ gathering in service systems, this paper investigates customer gathering levels by evaluating how long a virtual customer overlaps with others and how many people the customer encounters in queueing systems. Existing studies mainly characterize customer overlap in stationary systems, which are not applicable to more practical scenarios with dynamic and time-varying arrivals. Current computational methods dealing with time-varying systems only consider metrics like queue length and waiting time distributions. However, unlike these classic queueing metrics, both overlap time and the number of encounters depend on the current queue status and future arrivals during the sojourn time. The inherent randomness in both sojourn time and arrivals presents challenges in accurately approximating these overlap metrics. To address this, we propose a framework that uses the fluid limit to approximate the expected overlap time and number in time-varying queueing systems. We also prove the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6846_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^1\)</EquationSource> </InlineEquation>-convergence of the scaled processes, which differs from previous studies that focused on proving almost sure convergence for classic metrics. Simulation experiments show both the effectiveness and limitations of the fluid approximation. To mitigate the limitations of the fluid limit, we apply an adjusted limit approach for improved accuracy. Based on our analytical and numerical results, we further explore when customers are at the highest risk of overlap, revealing the fundamental differences between overlap metrics and traditional queueing metrics.</p>

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Overlap approximation schemes in time-varying many-server queues

  • 期刊论文

  • 发表日期：

    2025年9月17日

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  • Young Myoung Ko,
  • Jin Xu

摘要

Motivated by the need to manage customers’ gathering in service systems, this paper investigates customer gathering levels by evaluating how long a virtual customer overlaps with others and how many people the customer encounters in queueing systems. Existing studies mainly characterize customer overlap in stationary systems, which are not applicable to more practical scenarios with dynamic and time-varying arrivals. Current computational methods dealing with time-varying systems only consider metrics like queue length and waiting time distributions. However, unlike these classic queueing metrics, both overlap time and the number of encounters depend on the current queue status and future arrivals during the sojourn time. The inherent randomness in both sojourn time and arrivals presents challenges in accurately approximating these overlap metrics. To address this, we propose a framework that uses the fluid limit to approximate the expected overlap time and number in time-varying queueing systems. We also prove the \(L^1\) -convergence of the scaled processes, which differs from previous studies that focused on proving almost sure convergence for classic metrics. Simulation experiments show both the effectiveness and limitations of the fluid approximation. To mitigate the limitations of the fluid limit, we apply an adjusted limit approach for improved accuracy. Based on our analytical and numerical results, we further explore when customers are at the highest risk of overlap, revealing the fundamental differences between overlap metrics and traditional queueing metrics.