Heterogeneous server retrial queueing model with feedback and working vacation using artificial bee colony optimization algorithm

Kothandaraman Divya   ; Kandaiyan Indhira

Ural mathematical journal, Tome 9 (2023) no. 2, pp. 60-76

Résumé

This research delves into the dynamics of a retrial queueing system featuring heterogeneous servers with intermittent availability, incorporating feedback and working vacation mechanisms. Employing a matrix geometric approach, this study establishes the steady-state probability distribution for the queue size in this complex heterogeneous service model. Additionally, a range of system performance metrics is developed, alongside the formulation of a cost function to evaluate decision variable optimization within the service system. The Artificial Bee Colony (ABC) optimization algorithm is harnessed to determine service rates that minimize the overall cost. This work includes numerical examples and sensitivity analyses to validate the model's effectiveness. Also, a comparison between the numerical findings and the neuro-fuzzy results has been examined by the adaptive neuro fuzzy interface system (ANFIS).

Détail

Keywords: retrial queue, working vacation, MGA, ANFIS, ABC optimization.

Bibliographie

[1] Agarwal N. N., Some Problems in the Theory of Reliability and Queues, Ph.D. Thesis., Kurukshetra University, Kurukshetra, India, 1965

[2] Ahuja A., Jain A., Jain M., “Transient analysis and ANFIS computing of unreliable single server queueing model with multiple stage service and functioning vacation”, Math. Comput. Simulation, 192 (2022), 464–490 | DOI | MR

[3] Bouchentouf A. A., Kadi M., Rabhi A., “Analysis of two heterogeneous server queueing model with balking, reneging and feedback”, Math. Sci. Appl E-Notes, 2:2 (2014), 10–21 | DOI | Zbl

[4] Chakravarthy S. R., “Analysis of a queueing model with MAP arrivals and heterogeneous phase-type group services”, Mathematics, 10:19 (2022), 3575 | DOI

[5] Divya K., Indhira K., “Analysis of a heterogeneous queuing model with intermittently obtainable servers under a hybrid vacation schedule”, Symmetry, 15:7 (2023), 1304 | DOI

[6] Divya K., Indhira K., “Performance analysis and ANFIS computing of an unreliable Markovian feedback queuing model under a hybrid vacation policy”, Math. Comput. Simulation, 218 (2024), 403–419 | DOI | MR

[7] Jain M., Jain A., “Working vacations queueing model with multiple types of server breakdowns”, Appl. Math. Model., 34:1 (2010), 1–13 | DOI | MR | Zbl

[8] Neuts M. F., “Matrix-Geometric solutions in stochastic models”, DGOR. Operations Research Proceedings, v. 1983, eds. Steckhan H. and al., Springer, Berlin, Heidelberg, 1984, 425 pp. | DOI

[9] Krishnamoorthy A., Sreenivasan C., “An $M/M/2$ queueing system with heterogeneous servers including one with working vacation”, Int. J. Stoch. Anal., 2012 (2012), 145867 | DOI | Zbl

[10] Kumar A., Jain M., “Cost optimization of an unreliable server queue with two stage service process under hybrid vacation policy”, Math. Comput. Simulation, 204 (2023), 259–281 | DOI | MR

[11] Kumar R., Sharma S. K., “Two heterogeneous server Markovian queueing model with discouraged arrivals, reneging and retention of reneged customers”, Int. J. Oper. Res., 11:2 (2014), 64–68 | MR

[12] Leemans H. E., The Two-Class Two-Server Queueing Model with Nonpreemptive Heterogeneous Priority Structures, Ph.D. Thesis, Katholieke Universiteit Leuven, Leuven, 1998

[13] Servi L. D., Finn S. G., “$M/M/1$ queues with working vacations $(M/M/1/WV)$”, Perform. Eval., 50:1 (2002), 41–52 | DOI

[14] Morse P. M., Queues, Inventories and Maintenance: The Analysis of Operational Systems with Variable Demand and Supply, Courier Corporation, Mineola, New York, 2004, 202 pp.

[15] Latouche G., Ramaswami V., Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series Stat. Appl. Math., SIAM, Philadelphia, Pennsylvania; ASA, Alexandria, Virginia, 1999, 334 pp. | DOI | Zbl

[16] Seenivasan M., Senthilkumar R., Subasri K. S., “$M/M/2$ heterogeneous queueing system having unreliable server with catastrophes and restoration”, Mater. Today: Proc., 51:8 (2022), 2332–2338 | DOI

[17] Sethi R., Jain M., Meena R. K., Garg D., “Cost optimization and ANFIS computing of an unreliable $M/M/1$ queueing system with customers' impatience under $N$-policy”, Int. J. Appl. Comput. Math., 6 (2020), 51, 1–14 | DOI

[18] Sanga S. S., Jain M., “Cost optimization and ANFIS computing for admission control of $M/M/1/K$ queue with general retrial times and discouragement”, Appl. Math. Comput., 363 (2019), 124624 | DOI | MR | Zbl

[19] Sharda., “A queuing problem with intermittently available server and arrivals and departures in batches of variable size”, ZAMM, 48 (1968), 471–476 | DOI | MR | Zbl

[20] Stojčić M., Pamučar D., Mahmutagić E., Stević Ž., “Development of an ANFIS Model for the Optimization of a Queuing System in Warehouses”, Information, 9:10 (2018), 240, 1–20 | DOI

[21] Sudhesh R., Azhagappan A., Dharmaraja S., “Transient analysis of $M/M/1$ queue with working vacation, heterogeneous service and customers' impatience”, RAIRO-Oper. Res., 51:3 (2017), 591–606 | DOI | MR | Zbl

[22] Thakur S., Jain A., Jain M., “ANFIS and cost optimization for Markovian queue with operational vacation”, Int. J. Math. Eng. Management Sci., 6:3 (2021), 894–910 | DOI

[23] Vijaya Laxmi P., Jyothsna K., “Balking and reneging multiple working vacations queue with heterogeneous servers”, J. Math. Model. Algor., 14 (2015), 267–285 | DOI | Zbl

[24] Wu C.-H., Yang D.-Y., “Bi-objective optimization of a queueing model with two-phase heterogeneous service”, Comput. Oper. Res., 130 (2021), 105230 | DOI | Zbl

[25] Yohapriyadharsini R. S., Suvitha V., “Multi-server Markovian heterogeneous arrivals queue with two kinds of working vacations and impatient customers”, Yugosl. J. Oper. Res., 33:4 (2023), 643–666 | DOI | MR

[26] Zadeh L. A., “The concept of a linguistic variable and its application to approximate reasoning–I”, Inform. Sci., 8:3 (1975), 199–249 | DOI | Zbl