Analytical and Computational Study of Bi-level Vacation Policy for Transient Markovian Queue with State-Dependent Rates, Waiting Server, and Discouragement

doi:10.1007/s42967-024-00454-2

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (2): 507-532.doi: 10.1007/s42967-024-00454-2

Analytical and Computational Study of Bi-level Vacation Policy for Transient Markovian Queue with State-Dependent Rates, Waiting Server, and Discouragement

Mayank Singh, Madhu Jain, Vijay Pratap Singh

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

收稿日期:2023-12-16 修回日期:2024-08-06 出版日期:2026-04-07 发布日期:2026-04-07
通讯作者: Madhu Jain,E-mail:madhu.jain@ma.iitr.ac.in E-mail:madhu.jain@ma.iitr.ac.in
基金资助:
This article is funded by the University Grants Commission, India for awarding JRF/SRF (grant code 11-04-2016-429688, Mayank Singh).

Analytical and Computational Study of Bi-level Vacation Policy for Transient Markovian Queue with State-Dependent Rates, Waiting Server, and Discouragement

Mayank Singh, Madhu Jain, Vijay Pratap Singh

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

Received:2023-12-16 Revised:2024-08-06 Online:2026-04-07 Published:2026-04-07
Contact: Madhu Jain,E-mail:madhu.jain@ma.iitr.ac.in E-mail:madhu.jain@ma.iitr.ac.in
Supported by:
This article is funded by the University Grants Commission, India for awarding JRF/SRF (grant code 11-04-2016-429688, Mayank Singh).

摘要/Abstract

摘要： In certain queueing circumstances, a bi-level vacation policy according to which the server can switch over from the working vacation (WV) to the complete vacation (CV) has been noticed. In the WV (CV) mode, customers may exhibit reneging behavior due to the slow (absence of) service. This study examines the queueing metrics of a Markovian queueing system incorporating the concepts of the bi-level hybrid vacation strategy, state-dependent rates, discouraged customers, and waiting servers. The continued fraction, the probability generating function (PGF), and the confluent hypergeometric function are used to derive the transient queue size distributions. As a special case of the transient results, we derive the steady-state results. The analytical results for the average system size, variance, average reneging rate, system throughput, etc. are established. Furthermore, the cost function framed is optimized to provide the optimal decision parameters and respective minimum cost. Numerical simulation and parameter sensitivity have been performed by taking illustrations.

关键词: Transient, Markov queue, Bi-level vacation, Reneged customers, Continued fraction, Queue length

Abstract: In certain queueing circumstances, a bi-level vacation policy according to which the server can switch over from the working vacation (WV) to the complete vacation (CV) has been noticed. In the WV (CV) mode, customers may exhibit reneging behavior due to the slow (absence of) service. This study examines the queueing metrics of a Markovian queueing system incorporating the concepts of the bi-level hybrid vacation strategy, state-dependent rates, discouraged customers, and waiting servers. The continued fraction, the probability generating function (PGF), and the confluent hypergeometric function are used to derive the transient queue size distributions. As a special case of the transient results, we derive the steady-state results. The analytical results for the average system size, variance, average reneging rate, system throughput, etc. are established. Furthermore, the cost function framed is optimized to provide the optimal decision parameters and respective minimum cost. Numerical simulation and parameter sensitivity have been performed by taking illustrations.

Key words: Transient, Markov queue, Bi-level vacation, Reneged customers, Continued fraction, Queue length

中图分类号:

Mayank Singh, Madhu Jain, Vijay Pratap Singh. Analytical and Computational Study of Bi-level Vacation Policy for Transient Markovian Queue with State-Dependent Rates, Waiting Server, and Discouragement[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 507-532.

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Analytical and Computational Study of Bi-level Vacation Policy for Transient Markovian Queue with State-Dependent Rates, Waiting Server, and Discouragement

Analytical and Computational Study of Bi-level Vacation Policy for Transient Markovian Queue with State-Dependent Rates, Waiting Server, and Discouragement

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