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<p>The paper investigates the transient behavior and cost optimization of the <i>M</i>/<i>M</i>/1/<i>K</i> retrial queueing model incorporating customers balking, server breakdowns, and an <i>N</i>-policy. The model explores a single-server queue with finite capacity <i>K</i>, where customers may opt out of the system due to congestion without joining the queue. Furthermore, server breakdowns can occur, resulting in temporary service interruptions. Repair activities commence only when the total number of customers reaches a predefined threshold, denoted as <i>N</i>. If the server is occupied, arriving customers have the choice to wait in a virtual space known as the retrial orbit, from which they can later attempt to receive service again. The mathematical analysis of the model is conducted using the matrix method, solving the system of differential-difference equations. Expressions for various performance indices, including the number of customers in the system, throughput, and long-run probabilities, are derived and their behaviors are analyzed across different time intervals and input parameters. Moreover, a nonlinear cost function is formulated and minimized using the quasi-Newton method.</p>

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A Study on the Transient Behavior of the M/M/1/K Retrial Queue with N-Policy and Server Breakdown

  • 期刊论文

  • 发表日期：

    2026年6月3日

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  • Sudeep Singh Sanga,
  • Vijaykumar Panchal

摘要

The paper investigates the transient behavior and cost optimization of the M/M/1/K retrial queueing model incorporating customers balking, server breakdowns, and an N-policy. The model explores a single-server queue with finite capacity K, where customers may opt out of the system due to congestion without joining the queue. Furthermore, server breakdowns can occur, resulting in temporary service interruptions. Repair activities commence only when the total number of customers reaches a predefined threshold, denoted as N. If the server is occupied, arriving customers have the choice to wait in a virtual space known as the retrial orbit, from which they can later attempt to receive service again. The mathematical analysis of the model is conducted using the matrix method, solving the system of differential-difference equations. Expressions for various performance indices, including the number of customers in the system, throughput, and long-run probabilities, are derived and their behaviors are analyzed across different time intervals and input parameters. Moreover, a nonlinear cost function is formulated and minimized using the quasi-Newton method.