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Convexity Conditions for Optimizing a Single Server Discrete-time Queueing System under a Randomized Cutoff Policy

Author

Listed:
  • Shweta Upadhyaya

    (Department of Mathematics, School of Computer Science Engineering and Technology, Bennett University)

  • Divya Agarwal

    (Amity Institute of Applied Sciences, Amity University)

  • Shree Vaishnawi

    (Amity Institute of Applied Sciences, Amity University)

Abstract

This research aims to investigate convexity conditions for a single server discrete-time queueing system with setback (disaster). The system operates under a randomized threshold policy $$\left(p,\;N\right)$$ p , N , where the server activates with probability $$p$$ p when queue length reaches threshold $$N$$ N , or remains idle with probability $$1-p$$ 1 - p , providing flexible control over system congestion and resource utilization. This policy is particularly important in systems prone to sudden disruptions, as it helps optimize service efficiency while managing system recovery after setbacks. First, we perform the convexity analysis analytically for the discrete parameter $$N$$ N . Then the optimal queue length for the best value of $$N$$ N is determined. Also, as the optimization problem for finding the optimal value of continuous parameter $$p$$ p is a linear fractional programming problem thus Charnes and Cooper method is used to get the best value of $$p$$ p . Furthermore, by constructing a cost function and using the direct search method and firefly algorithm, the minimum cost is estimated. The goal of this work is to show how convexity in a discrete-time work frame can provide fresh perspectives on existing problems and lead to significantly simpler analyses and algorithm modifications. Also we compare analytical results with that of ANFIS (Adaptive Neuro-Fuzzy Inference System) results which validate our findings.

Suggested Citation

  • Shweta Upadhyaya & Divya Agarwal & Shree Vaishnawi, 2025. "Convexity Conditions for Optimizing a Single Server Discrete-time Queueing System under a Randomized Cutoff Policy," Methodology and Computing in Applied Probability, Springer, vol. 27(3), pages 1-26, September.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:3:d:10.1007_s11009-025-10183-5
    DOI: 10.1007/s11009-025-10183-5
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    References listed on IDEAS

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    1. Sung J. Kim & Nam K. Kim & Hyun-Min Park & Kyung Chul Chae & Dae-Eun Lim, 2013. "On the Discrete-Time Queues under -Policy with Single and Multiple Vacations," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-6, December.
    2. Sung J. Kim & Nam K. Kim & Hyun-Min Park & Kyung Chul Chae & Dae-Eun Lim, 2013. "On the Discrete‐Time GeoX/G/1 Queues under N‐Policy with Single and Multiple Vacations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    3. Shweta Upadhyaya & Richa Sharma & Divya Agarwal & Geetika Malik, 2023. "Convexity analysis and cost optimization of a retrial queue with Bernoulli vacation and delayed phase mending," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 14(5), pages 1671-1690, October.
    4. Schaible, Siegfried & Ibaraki, Toshidide, 1983. "Fractional programming," European Journal of Operational Research, Elsevier, vol. 12(4), pages 325-338, April.
    5. Radhika Agarwal & Shweta Upadhyaya & Divya Agarwal & Sumit Kumar, 2023. "Cost optimality of an erratic Geo X / G /1 retrial queue under J-vacation scheme using nature inspired algorithms," International Journal of Industrial and Systems Engineering, Inderscience Enterprises Ltd, vol. 44(1), pages 1-33.
    6. A. Charnes & W. W. Cooper, 1962. "Programming with linear fractional functionals," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 9(3‐4), pages 181-186, September.
    7. Mustafa Demircioglu & Herwig Bruneel & Sabine Wittevrongel, 2021. "Analysis of a Discrete-Time Queueing Model with Disasters," Mathematics, MDPI, vol. 9(24), pages 1-22, December.
    8. Shweta Upadhyaya & Geetika Malik & Richa Sharma, 2022. "Neuro-fuzzy computing and optimisation results for batch discrete time retrial queue," International Journal of Mathematics in Operational Research, Inderscience Enterprises Ltd, vol. 23(1), pages 119-146.
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