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<p>Integrated queueing-inventory systems present unique analytical challenges due to the coupled stochastic behavior of customer arrivals, service times, and stock replenishment. We study an <i>M</i>/<i>M</i>/1 queueing-inventory system with a finite buffer and dual replenishment policies, <i>normal</i> and <i>emergency</i>, designed to manage inventory effectively under stochastic customer demand. Customer arrivals follow a Poisson process, while service and replenishment times are exponentially distributed. Customers employ threshold-based strategies to decide whether to join the system and whether to purchase after joining, depending on the level of system observability (fully or partially observable settings). Inventory is replenished normally to a level <i>Q</i> when it drops to a threshold <i>r</i>. In case of failure to replenish in time, an <i>emergency</i> replenishment is triggered when the inventory hits a threshold lower than <i>r</i>, restoring the stock to a threshold <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Q_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> to mitigate customer loss. We derive a product-form steady-state distribution of the system state and identify equilibrium customer strategies under selfish behavior. The service provider’s revenue is optimized, and socially optimal strategies are also derived. We measure the efficiency loss due to individually rational behavior using the <i>price of anarchy</i> and the <i>price of conservatism</i>. Numerical experiments illustrate the effects of profit-driven versus socially optimal strategies, and how reorder thresholds and replenishment quantities impact overall system performance.</p>

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Equilibrium Analysis of an Observable Queueing-Inventory System with Normal and Emergency Replenishment Policies

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • Salini K,
  • Manikandan Rangaswamy,
  • Bo Li

摘要

Integrated queueing-inventory systems present unique analytical challenges due to the coupled stochastic behavior of customer arrivals, service times, and stock replenishment. We study an M/M/1 queueing-inventory system with a finite buffer and dual replenishment policies, normal and emergency, designed to manage inventory effectively under stochastic customer demand. Customer arrivals follow a Poisson process, while service and replenishment times are exponentially distributed. Customers employ threshold-based strategies to decide whether to join the system and whether to purchase after joining, depending on the level of system observability (fully or partially observable settings). Inventory is replenished normally to a level Q when it drops to a threshold r. In case of failure to replenish in time, an emergency replenishment is triggered when the inventory hits a threshold lower than r, restoring the stock to a threshold \(Q_n\) Q n to mitigate customer loss. We derive a product-form steady-state distribution of the system state and identify equilibrium customer strategies under selfish behavior. The service provider’s revenue is optimized, and socially optimal strategies are also derived. We measure the efficiency loss due to individually rational behavior using the price of anarchy and the price of conservatism. Numerical experiments illustrate the effects of profit-driven versus socially optimal strategies, and how reorder thresholds and replenishment quantities impact overall system performance.