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Analysis of a Batch Arrival, Batch Service Queuing-Inventory System with Processing of Inventory While on Vacation

Author

Listed:
  • Achyutha Krishnamoorthy

    (Centre for Research in Mathematics, C.M.S. College, Kottayam 686001, India)

  • Anu Nuthan Joshua

    (Department of Mathematics, Union Christian College, Aluva 683102, India
    Working for Doctoral Degree at Department of Mathematics, Cochin University of Science and Technology, Cochin-22.)

  • Dmitry Kozyrev

    (Applied Probability and Informatics Department, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russia
    V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Street, 117997 Moscow, Russia)

Abstract

A single-server queuing-inventory system in which arrivals are governed by a batch Markovian arrival process and successive arrival batch sizes form a finite first-order Markov chain is considered in this paper. Service is provided in batches according to a batch Markovian service process, with consecutive service batch sizes forming a finite first-order Markov chain. A service starts for the next batch on completion of the current service, provided that inventory is available at that epoch; otherwise, there will be a delay in starting the next service. When the service of a batch is completed, the inventory decreases by 1 unit, irrespective of batch size. A control policy in which the server goes on vacation when a service process is frozen until a quorum can initiate the next batch service is proposed to ensure idle-time utilization. During the vacation, the server produces inventory (items) for future services until it hits a specified level L or until the number of customers in the system reaches a maximum service batch size N , with whichever occurring first. In the former case, a server stays idle once the processed inventory level reaches L until the number of customers reaches (or even exceeds because of batch arrival) a maximum service batch size N . The time required for processing one unit of inventory follows a phase-type distribution. In this paper, the steady-state probability vector of this infinite system is computed. The distributions of inventory processing time in a vacation cycle, idle time in a vacation cycle, and vacation cycle length are found. The effect of correlation in successive inter-arrival times and service times on performance measures for such a queuing system is illustrated with a numerical example. An optimization problem is considered. The proposed system is then compared with a queuing-inventory system without the Markov-dependent assumption on successive arrivals as well as service batch sizes using numerical examples.

Suggested Citation

  • Achyutha Krishnamoorthy & Anu Nuthan Joshua & Dmitry Kozyrev, 2021. "Analysis of a Batch Arrival, Batch Service Queuing-Inventory System with Processing of Inventory While on Vacation," Mathematics, MDPI, vol. 9(4), pages 1-29, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:419-:d:502974
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    References listed on IDEAS

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    1. Yonatan Levy & Uri Yechiali, 1975. "Utilization of Idle Time in an M/G/1 Queueing System," Management Science, INFORMS, vol. 22(2), pages 202-211, October.
    2. Naishuo Tian & Zhe George Zhang, 2006. "Vacation Queueing Models Theory and Applications," International Series in Operations Research and Management Science, Springer, number 978-0-387-33723-4, April.
    3. S. R. Chakravarthy & Arunava Maity & U. C. Gupta, 2017. "An ‘(s, S)’ inventory in a queueing system with batch service facility," Annals of Operations Research, Springer, vol. 258(2), pages 263-283, November.
    4. Naishuo Tian & Zhe George Zhang, 2006. "Applications of Vacation Models," International Series in Operations Research & Management Science, in: Vacation Queueing Models Theory and Applications, chapter 0, pages 343-358, Springer.
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    2. Sujit Kumar Samanta & Kousik Das, 2023. "Detailed Analytical and Computational Studies of D-BMAP/D-BMSP/1 Queueing System," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-37, March.

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