Performance and economic analysis of an unreliable single-server queue with general retrial times and varied customer patience levels

Abstract

This paper presents a comprehensive mathematical analysis of an unreliable single-server retrial queue with general retrial times, serving two types of customer arrivals: high-patience and low-patience customers. Customers arrive in the system following two Poisson processes with different service rates. In addition, the model incorporates essential features such as service times, reserved times, and repair times, all following general distributions. The proposed model has practical applications in diverse domains, including healthcare systems, web traffic management, and call centers. Using the supplementary variable technique, we carry out an extensive analysis of the model. This approach allows us to derive the ergodicity condition for this Markov chain and compute its stationary distribution. The main performance measures of the system are expressed through the stationary state probabilities. Numerical illustrations are presented. Finally, we conduct an economic study to assess the impact of various system parameters on performance measures and total cost, offering a visual overview of the system's effectiveness and profitability. A comparative analysis with existing models shows how our approach generalizes traditional retrial queue models, which typically consider a single type of customer arrival, by considering two distinct customer classes. This contributes to the advancement of queueing theory and provides insight into optimizing real-world systems.

Keywords

• Breakdowns and repairs
• cost model
• retrial queues
• performance measures
• server state probabilities
• service orbit

References

1. [1] A. Aissani, F. Lounis, D. Hamadouche and S. Taleb, Analysis of customers’ impatience in a repairable retrial queue under postponed preventive actions, American Journal of Mathematical and Management Sciences, 38 (2), 125-150, 2019, https://doi.org/10.1080/01966324.2018.1486763.
2. [2] L.M. Alem, M. Boualem and D. Aïssani, Bounds of the stationary distribution in $M/G/1$ retrial queue with two-way communication and n types of outgoing calls, Yugoslav Journal of Operations Research, 29 (3), 375-39, 2019, https://doi.org/ 10.2298/YJOR180715012A.
3. [3] L.M. Alem, M. Boualem and D. Aïssani, Stochastic comparison bounds for an $M_1, M_2/G_1, G_2/1$ retrial queue with two way communication, Hacettepe Journal of Mathematics and Statistics, 48 (4), 1185-1200, 2019, https://dergipark.org.tr/ en/pub/hujms/issue/47862/604504.
4. [4] J. Artalejo and A. Gomez-Corral, Retrial queueing systems: A Computational Approach, Springer-Verlag, Berlin, 2008, https://api.semanticscholar.org/ CorpusID:60225921.
5. [5] G. Ayyappan and P. Thamizhselvi, Transient analysis of $M^{[X_1]}, M^{[X_2]}/G_1, G_2/1$ retrial queueing system with priority services, working vacations and vacation interruption, emergency vacation, negative arrival and delayed repair, International Journal of Applied and Computational Mathematics, 4 (2), 2018, https://doi.org/10.1007/ s40819-018-0509-7.
6. [6] G. Ayyappan and J. Udayageetha, Transient analysis of $M^{[X_1]}, M^{[X_2]}/G_1, G_2/1$ retrial queueing system with priority services, working breakdown, start up/close down time, Bernoulli vacation, reneging and balking, Pakistan Journal of Statistics and Operation Research, 16 (1), 203-216, 2020, https://doi.org/10.18187/pjsor.v16i1. 2181.
7. [7] M. Boualem, A. Bareche and M. Cherfaoui, Approximate controllability of stochastic bounds of stationary distribution of an $M/G/1$queue with repeated attempts and two phase service, International Journal of Management Science and Engineering Management, 14 (2), 79-85, 2018, https://api.semanticscholar.org/CorpusID: 125814082.
8. [8] A.A. Bouchentouf, M. Boualem, L. Yahiaoui and H. Ahmad, A multi-station unreliable machine model with working vacation policy and customer impatience, Quality Technology & Quantitative Management, 19 (6), 766-796, 2022, https://doi.org/ 10.1080/16843703.2022.2054088.

9. [9] A.A. Bouchentouf, M. Cherfaoui and M. Boualem, Performance and economic analysis of a single server feedback queueing model with vacation and impatient customers, Opsearch, 56 (1), 300-323, 2019, https://doi.org/10.1007/s12597-019-00357-4.
10. [10] A.A. Bouchentouf, M. Cherfaoui and M. Boualem, Analysis and performance evaluation of Markovian feedback multi-server queueing model with vacation and impatience, American Journal of Mathematical and Management Sciences, 40, 261-282, 2021, https://doi.org/10.1080/01966324.2020.1842271.
11. [11] M. Cherfaoui, A.A. Bouchentouf and M. Boualem, Modeling and simulation of Bernoulli feedback queue with general customers impatience under variant vacation policy, International Journal of Operational Research, 46, 451-480, 2023, https: //doi.org/10.1504/ijor.2023.129959.
12. [12] G. Choudhury and M. Deka, A batch arrival unreliable server delaying repair queue with two phases of service and Bernoulli vacation under multiple vacation policy, Quality Technology & Quantitative Management, 15 (2), 157-186, 2018, https:// doi.org/10.1080/16843703.2016.1208934.
13. [13] A. Dehimi, M. Boualem, A.A. Bouchentouf, S. Ziani and L. Berdjoudj, Analytical and computational aspects of a multi-server queue with impatience under differentiated working Vacations policy, Reliability: Theory & Applications 19, 3 (79), 393407, 2024, https://doi.org/10.24412/1932-2321-2024-379-393-407.
14. [14] S. Dhar, L.B. Mahanta and K.K. Das, Estimation of the waiting time of patients in a hospital with simple Markovian model using order statistics, Hacettepe Journal of Mathematics and Statistics, 48 (1), 274-289, 2019, https://doi.org/10.15672/ HJMS.2018.607.
15. [15] A. Dudin, O. Dudina, S. Dudin and K. Samouylov, Analysis of single-server multiclass queue with unreliable service, batch correlated arrivals, customers impatience, and dynamical change of priorities, Mathematics, 9 (11), 1257, 2021, https://doi. org/10.3390/math9111257.
16. [16] D. Fiems, Retrial queues with constant retrial times, Queueing Systems, 103 (3/4), 347-365, 2023, https://doi.org/10.1007/s11134-022-09866-4.
17. [17] S. Gao, A preemptive priority retrial queue with two classes of customers and general retrial times, Operational Research, 15 (2), 233-251, 2015, https://doi.org/10. 1007/s12351-015-0175-z.
18. [18] H. Gao, J. Zhang and X. Wang, Analysis of a retrial queue with two-type breakdowns and delayed repairs, IEEE Access, 8, 172428-172442, 2020, https://doi.org/10. 1109/ACCESS.2020.3023191.
19. [19] H. Hablal, N. Touche, L. Alem, A.A. Bouchentouf and M. Boualem, Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system, Hacettepe Journal of Mathematics and Statistics, 52 (5), 1438-1460, 2023, https://doi.org/10.15672/hujms.1183966.
20. [20] D. Hamadouche, A. Aissani, F. Lounis. On the asymptotic behaviour of an unreliable M/G/1 retrial queue with impatience, Authorea, 2024, https://doi.org/10.22541/ au.170668021.12989057/v1.
21. [21] K. C. Hariom, Sharma, K. Singh and D. Singh, Analysis of an inventory model for time-dependent linear demand rate three levels of production with shortage, International Journal of Professional Business Review, 9 (4), 2024, https://doi.org/10. 26668/businessreview/2024.v9i4.4579.
22. [22] B. Jagannathan and N. Sivasubramaniam, Bulk arrival queue with unreliable server, balking and modified Bernoulli vacation, Hacettepe Journal of Mathematics and Statistics, 53 (1), 289-304, 2024, https://doi.org/10.15672/hujms.1181711.
23. [23] M. Jain and A. Bhagat, $M^X/G/1$ retrial vacation queue for multi-optional services, phase repair and reneging, Quality Technology & Quantitative Management, 13, 263- 288, 2016, https://doi.org/10.1080/16843703.2016.1189025.
24. [24] B. Kim and J. Kim, Waiting time distributions in an $M/G/1$ retrial queue with two classes of customers, Annals of Operations Research, 252 (1), 121-134, 2017, https://doi.org/10.1007/s10479-015-1979-1.
25. [25] V. Klimenok, A. Dudin, O. Dudina and I. Kochetkova, Queueing system with two types of customers and dynamic change of a priority, Mathematics, 8 (5), 824, 2020, https://doi.org/10.3390/math8050824.
26. [26] B. Krishna Kumar, R. Rukmani, A. Thanikachalam and V. Kanakasabapathi, Performance analysis of retrial queue with server subject to two types of breakdowns and repairs, Operational Research, 18, 521-559, 2018, https://doi.org/10.1007/ s12351-016-0275-4.
27. [27] A. Kumar, M. Boualem and A.A. Bouchentouf, Optimal analysis of machine interference problem with standby, random switching failure, vacation interruption, and synchronized reneging, In Applications of Advanced Optimization Techniques in Industrial Engineering, 155-168, 2022, https://doi.org/10.1201/9781003089636-10.
28. [28] S.K. Lee, S. Dudin, O. Dudina, C.S. Kim and A. Klimenok, A priority queue with many customer types, correlated arrivals, and changing priorities, Mathematics, 8, 1292, 2020, https://doi.org/10.3390/math8081292.
29. [29] T. Li and L. Zhang, An $M/G/1$ retrial G-queue with general retrial times and working breakdowns, Mathematical and Computational Applications, 22, 15, 2017, https: //doi.org/10.3390/mca22010015.
30. [30] S.P. Madheswari, B.K. Kumar and P. Suganthi, Analysis of M/G/1 retrial queues with second optional service and customer balking under two types of Bernoulli vacation schedule, RAIRO-Operations Research, 53 (2), 415-443, 2019, https://doi.org/10. 1051/ro/2017029.
31. [31] S. Mahanta, N. Kumar and G. Choudhury, Study of a two types of general heterogeneous service queueing system in a single server with optional repeated service and feedback queue, Hacettepe Journal of Mathematics and Statistics, 53, 3, 851-878, 2024, https://doi.org/10.15672/hujms.1312795.
32. [32] A. Melikov, S. Aliyeva, J. Sztrik, Retrial queues with unreliable servers and delayed feedback, Mathematics, 9 (19), 2415, 2021, https://doi.org/10.3390/ math9192415.
33. [33] S. Muthusamy, N. Devadoss and S.I. Ammar, Reliability and optimization measures of retrial queue with different classes of customers under a working vacation schedule, Discrete Dynamics in Nature and Society, 2022, https://doi.org/10.1155/2022/ 6806104.
34. [34] D. Singh, Production inventory model of deteriorating items with holding cost, stock, and selling price with backlog, International Journal of Mathematics in Operational Research, 14 (2), 290-305, 2019, https://doi.org/10.1504/IJMOR.2019.097760.
35. [35] D. Singh, M.G. Alharbi, A. Jayswal and A. A. Shaikh, Analysis of inventory model for quadratic demand with three levels of production, Intelligent Automation & Soft Computing, 32 (1), 167-182, 2022, https://doi.org/10.32604/iasc.2022.021815.
36. [36] D. Singh, A. Jayswal, M. G. Alharbi and A. A. Shaikh, An investigation of a supply chain model for coordination of finished products and raw materials in a production system under different situations, Sustainability, 13 (22), 12601, 2021, https://doi. org/10.3390/su132212601.
37. [37] J. Sztrik, A. Tóth, E. Danilyuk, S. Moiseeva, Analysis of retrial queueing system M/G/1 with impatient customers, collisions and unreliable server using simulation, 1391, Communications in Computer and Information Science, 291-303, 2021, https: //doi.org/10.1007/978-3-030-72247-0_22.
38. [38] S. Taleb and A. Aissani, Preventive maintenance in an unreliable $M/G/1$ retrial queue with persistent and impatient customers, Annals of Operations Research, 247 (1), 291-317, 2016, https://doi.org/10.1007/s10479-016-2217-1.
39. [39] R. Tian and Y. Zhang, Analysis of $M/M/1$ queueing systems with negative customers and unreliable repairers, Communications in Statistics-Theory and Methods, 53 (21), 74917504, 2023, https://doi.org/10.1080/03610926.2023.2265000.
40. [40] A. Toth and J. Sztrik, Simulation of two-way communication retrial queueing systems with unreliable server and impatient customers in the orbit, Stochastic Modelingand Applied Research of Technology, 3, 45-50, 2023, https://doi.org/10.57753/ SMARTY.2023.39.42.006.
41. [41] X. Wu, P. Brill, M. Hlynka and J. Wang, An $M/G/1$ retrial queue with balking and retrials during service, International Journal of Operational Research, 1 (1/2), 30-51, 2005, https://doi.org/10.1504/IJOR.2005.007432.
42. [42] M. Yin, M. Yan, Y. Guo and M. Liu, Analysis of a pre-emptive two-priority queueing system with impatient customers and heterogeneous servers, Mathematics, 11, 3878, 2023, https://doi.org/10.3390/math11183878.
43. [43] Y. Zhang and J. Wang, Managing retrial queueing systems with boundedly rational customers, Journal of the Operational Research Society, 74 (3), 748-761, 2022, https: //doi.org/10.1080/01605682.2022.2053305.
44. [44] D. Zirem, M. Boualem, K. Adel-Aissanou and D. Aïssani, Analysis of a single server batch arrival unreliable queue with balking and general retrial time, Quality Technology & Quantitative Management, 16, 672-695, 2019, https://doi.org/10.1080/ 16843703.2018.1510359.