Unreliable retrial queueing system with working vacation

Bharathy Shanmugam; Mookkaiyah Chandran Saravanarajan; Bharathy Shanmugam; Mookkaiyah Chandran Saravanarajan

doi:10.3934/math.20231234

AIMS Mathematics

2023, Volume 8, Issue 10: 24196-24224. doi: 10.3934/math.20231234

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Unreliable retrial queueing system with working vacation

Department of Mathematics, Vellore Institute of Techonology, Vellore-632014, Tamil Nadu, India

Received: 26 April 2023 Revised: 16 June 2023 Accepted: 06 July 2023 Published: 11 August 2023
MSC : 60K25, 60K30, 90B22

This paper investigates an unreliable $ M/G(P_{1}, P_{2})/1 $ retrial queueing system with a woking vacation. An arriving customer successfully starts the first phase service with the probability $ \alpha $ or the server fails with the probability $ \bar{\alpha} $. Once failure happens, the serving customer is taken to the orbit. The failed server is taken for repair with some delay. Once the repair is comleted, the server is ready to provide service once again. In this background, we implemented the working vacation scenario. During working vacation, the service will be provided at a slower rate, rather than entirely stopping the service. The supplementary variable method was adopted to find the orbit and system lengths. Additionally, some unique results and numerical evaluations have been presented.
- retrial queues,
- starting failure,
- two phase service,
- working vacation,
- repair
Citation: Bharathy Shanmugam, Mookkaiyah Chandran Saravanarajan. Unreliable retrial queueing system with working vacation[J]. AIMS Mathematics, 2023, 8(10): 24196-24224. doi: 10.3934/math.20231234

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Abstract

This paper investigates an unreliable $ M/G(P_{1}, P_{2})/1 $ retrial queueing system with a woking vacation. An arriving customer successfully starts the first phase service with the probability $ \alpha $ or the server fails with the probability $ \bar{\alpha} $. Once failure happens, the serving customer is taken to the orbit. The failed server is taken for repair with some delay. Once the repair is comleted, the server is ready to provide service once again. In this background, we implemented the working vacation scenario. During working vacation, the service will be provided at a slower rate, rather than entirely stopping the service. The supplementary variable method was adopted to find the orbit and system lengths. Additionally, some unique results and numerical evaluations have been presented.

References

[1]	G. Falin, A survey of retrial queues, Queueing Syst., 7 (1990), 127–167.
[2]	G. Falin, J. G. Templeton, Retrial queues, CRC Press, 75 (1997).
[3]	J. Artelejo, A. Gómez-Corral, Retrial queueing systems, 2008.
[4]	A. Gómez-Corral, A bibliographical guide to the analysis of retrial queues through matrix analytic techniques, Ann. Oper. Res., 141 (2006), 163–191. https://doi.org/10.1007/s10479-006-5298-4 doi: 10.1007/s10479-006-5298-4
[5]	J. R. Artalejo, Accessible bibliography on retrial queues, progress in 2000–2009, Math. Comput. Model. Dyn. Syst., 51 (2010), 1071–1081. https://doi.org/10.1016/j.mcm.2009.12.011 doi: 10.1016/j.mcm.2009.12.011
[6]	B. Mohamed, Stochastic analysis of a single server unreliable queue with balking and general retrial time, Discrete Cont. Model. Appl. Comput. Sci., 28 (2020), 319–326. https://doi.org/10.22363/2658-4670-2020-28-4-319-326 doi: 10.22363/2658-4670-2020-28-4-319-326
[7]	S. Gao, J. Zhang, Strategic joining and pricing policies in a retrial queue with orbital search and its application to call centers, IEEE Access, 7 (2019), 129317–129326. https://doi.org/10.1109/ACCESS.2019.2940287 doi: 10.1109/ACCESS.2019.2940287
[8]	A. Mohammadi, M. R. S. Rad, An M/G/1 queueing model with k sequential heterogeneous service steps and vacations in the transient state, Qual. Technol. Quant. Manag., 19 (2022), 633–647. https://doi.org/10.1080/16843703.2021.1981529 doi: 10.1080/16843703.2021.1981529
[9]	J. Sztrik, A. Tóth, A. Pintér, Z. Bács, The effect of operation time of the server on the performance of finite-source retrial queues with two-way communications to the orbit, J. Math. Sci., 267 (2022), 196–204. https://doi.org/10.1007/s10958-022-06124-z doi: 10.1007/s10958-022-06124-z
[10]	M. Varalakshmi, P. Rajadurai, M. C. Saravanarajan, V. M. Chandrasekaran, An M/G/1 retrial queueing system with two phases of service, immediate Bernoulli feedbacks, single vacation and starting failures, Int. J. Math. Oper. Res., 9 (2016), 302–328. https://doi.org/10.1504/IJMOR.2016.078823 doi: 10.1504/IJMOR.2016.078823
[11]	J. Wang, F. Wang, J. Sztrik, A. Kuki, Finite source retrial queues with two phase service, Int. J. Oper. Res., 30 (2017), 421–440. https://doi.org/10.1504/IJOR.2017.087824 doi: 10.1504/IJOR.2017.087824
[12]	L. M. Alem, M. Boualem, D. Aissani, Bounds of the stationary distribution in M/G/1 retrial queue with two way communication and n types of outgoing calls and n type of outgoing call, J. Oper. Res., 29 (2019), 375–391. Available from: https://yujor.fon.bg.ac.rs/index.php/yujor/article/view/651/632.
[13]	M. Boualem, A. Bareche, M. Cherfaoui, Approximate controllability of stochastic bounds of stationary distribution of an M/G/1 queue with repeated attempts and two-phase service, Int. J. Manag. Sci. Eng. Manag., 14 (2019), 79–85. https://doi.org/10.1080/17509653.2018.1488634 doi: 10.1080/17509653.2018.1488634
[14]	J. C. Ke, T. H. Liu, D. Y. Yang, Retrial queues with starting failure and service interruption, IET Commun., 12 (2018), 1431–1437.
[15]	B. K. Kumar, S. P. Madheswari, A. Vijayakumar, The M/G/1 retrial queue with feedback and starting failures, Appl. Math. Model., 26 (2002), 1057–1075. https://doi.org/10.1016/S0307-904X(02)00061-6 doi: 10.1016/S0307-904X(02)00061-6
[16]	S. Lan, Y. Tang, An unreliable discrete-time retrial queue with probabilistic preemptive priority, balking customers and replacements of repair times, AIMS Math., 5 (2020), 4322–4344. http://dx.doi.org/10.3934/math.2020276 doi: 10.3934/math.2020276
[17]	P. Rajadurai, R. Madhumidha, M. Sundararaman, D. Narasimhan, M/G/1 retrial queueing system with working vacation and starting failure, In: fuzzy mathematical analysis and advances in computational mathematics, Springer, Singapore, 2022,181–188. https://doi.org/10.1007/978-981-19-0471-413
[18]	D. Arivudainambi, M. Gowsalya, Analysis of an M/G/1 retrial queue with bernoulli vacation, two types of service and starting failure, Int. J. Artif. Intell., 6 (2017), 222–249. https://doi.org/10.1504/IJAISC.2017.088889 doi: 10.1504/IJAISC.2017.088889
[19]	S. I. Ammar, P. Rajadurai, Performance analysis of preemptive priority retrial queueing system with disaster under working breakdown services, Symmetry, 11 (2019), 419. https://doi.org/10.3390/sym11030419 doi: 10.3390/sym11030419
[20]	G. Choudhury, J. C. Ke, A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair, Appl. Math. Model., 36 (2012), 255–269. https://doi.org/10.1016/j.apm.2011.05.047 doi: 10.1016/j.apm.2011.05.047
[21]	T. H. Liu, J. C. Ke, C. C. Kuo, F. M. Chang, On the retrial queue with imperfect coverage and delay reboot, RAIRO-Oper. Res., 55 (2021), S1229–S1248. https://doi.org/10.1051/ro/2020103 doi: 10.1051/ro/2020103
[22]	M. S. Kumar, K. Sohraby, K. Kim, Delay analysis of orderly reattempts in retrial queueing system with phase type retrial time, IEEE Commun. Lett., 17 (2013), 822–825. https://doi.org/10.1109/LCOMM.2013.040913.122572 doi: 10.1109/LCOMM.2013.040913.122572
[23]	H. Saggou, T. Lachemot, M. Ourbih-Tari, Performance measures of M/G/1 retrial queues with recurrent customers, breakdowns, and general delays, Commun. Stat., 46 (2017), 7998–8015. https://doi.org/10.1080/03610926.2016.1171352 doi: 10.1080/03610926.2016.1171352
[24]	J. Wang, Z. Wang, Y. Liu, Reducing delay in retrial queues by simultaneously differentiating service and retrial rates, Oper. Res., 68 (2020), 1648–1667. https://doi.org/10.1287/opre.2019.1933 doi: 10.1287/opre.2019.1933
[25]	P. Rajadurai, M. C. Saravanarajan, V. M. Chandrasekaran, A study on M/G/1 feedback retrial queue with subject to server breakdown and repair under multiple working vacation policy, Alex. Eng. J., 57 (2018), 947–962. https://doi.org/10.1016/j.aej.2017.01.002 doi: 10.1016/j.aej.2017.01.002
[26]	V. G. Kulkarni, B. D. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Syst., 1990,191–208. https://doi.org/10.1007/BF01158474 doi: 10.1007/BF01158474
[27]	S. Abdollahi, M. R. S. Rad, Reliability and sensitivity analysis of a batch arrival retrial queue with k-phase services, feedback, vacation, delay, repair and admission, Int. J. Reliab. Qual. Sa., 3 (2020), 27–40. https://doi.org/10.30699/IJRRS.3.2.4 doi: 10.30699/IJRRS.3.2.4
[28]	S. Gao, J. Zhang, X. Wang, Analysis of a retrial queue with two-type breakdowns and delayed repairs, IEEE Access, 8 (2020), 172428–172442. https://doi.org/10.1109/ACCESS.2020.3023191 doi: 10.1109/ACCESS.2020.3023191
[29]	A. A. Bouchentouf, L. Yahiaoui, On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption, Arab. J. Math., 6 (2017), 1–11. https://doi.org/10.1007/s40065-016-0161-1 doi: 10.1007/s40065-016-0161-1
[30]	S. Jeyakumar, B. Senthilnathan, Modelling and analysis of a bulk service queueing model with multiple working vacations and server breakdown, RAIRO Oper. Res., 51 (2017), 485–508.
[31]	P. Manoharan, S. Majid, Stationary analysis of a multiserver queue with multiple working vacation and impatient customers, Appl. Appl. Math., 12 (2017), 2. https://doi.org/10.1051/ro/2016037 doi: 10.1051/ro/2016037
[32]	P. B. Muruga, R. Vijaykrishnaraj, A bulk arrival retrial queue with starting failures and exponentially distributed multiple working vacation, J. Xi'an Univ. Archit. Technol., 7 (2020), 3080–3088.
[33]	P. Gupta, N. Kumar, Performance analysis of retrial queueing model with working vacation, interruption, waiting server, breakdown and repair, J. Sci. Res., 13 (2021), 833–844. https://doi.org/10.3329/jsr.v13i3.52546 doi: 10.3329/jsr.v13i3.52546
[34]	A. A. Bouchentouf, M. Boualem, L. Yahiaoui, H. Ahmad, A multi-station unreliable machine model with working vacation policy and customers' impatience, Qual. Technol. Quant. Manag., 19 (2022), 766–796. https://doi.org/10.1080/16843703.2022.2054088 doi: 10.1080/16843703.2022.2054088
[35]	Y. Zhang, J. Wang, Strategic joining and information disclosing in Markovian queues with an unreliable server and working vacations, Qual. Technol. Quant. Manag., 18 (2021), 298–325. https://doi.org/10.1080/16843703.2020.1809062 doi: 10.1080/16843703.2020.1809062
[36]	M. Gowsalya, D. Arivudainambi, Stochastic analysis of an M/G/1 retrial queue subject to working vacation and starting failure, AIP Conf. Proc., 2095 (2019).
[37]	D. Arivudainambi, P. Godhandaraman, P. Rajadurai, Performance analysis of a single server retrial queue with working vacation, Oper. Res., 51 (2014), 434–462. https://doi.org/10.1007/s12597-013-0154-1 doi: 10.1007/s12597-013-0154-1
[38]	M. Zhang, Z. Hou, M/G/1 queue with single working vacation, J. Appl. Math. Comput., 39 (2012), 221–234. https://doi.org/10.1007/s12190-011-0532-x doi: 10.1007/s12190-011-0532-x
[39]	J. F. Shortle, J. M. Thompson, D. Gross, C. M. Harris, Fundamentals of queueing theory, John Wiley and Sons, 399 (2018).
[40]	L. I. Sennott, P. A. Humblet, R. L. Tweedie, Mean drifts and the non-ergodicity of Markov chains, Oper. Res., 4 (1983), 783–789. https://doi.org/10.1287/opre.31.4.783 doi: 10.1287/opre.31.4.783

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