Analysing the Performance of  Queuing Model with the Busy Period Breakdown

Authors

  • Lidiya P Department of Mathematics, Nirmala College for Women, Bharathiar University, Coimbatore, Tamil Nadu, India. https://orcid.org/0009-0002-8314-7405
  • K Julia Rose Mary Department of Mathematics, Nirmala College for Women, Bharathiar University, Coimbatore, Tamil Nadu, India.

DOI:

https://doi.org/10.21123/bsj.2024.9155

Keywords:

Breakdown, busy state, Idle, Multiple working vacation, Working state

Abstract

This paper aims to analyze the M/M(a,b)/1 multiple working vacations queuing model with a breakdown. Instead of the server being fully idle during the vacation period, the server serves at a different rate during multiple working vacations. The system has only one server, and the service rate varies depending on the arrival state. Customers’ enter the system to get service with parameter λ_v following the Poisson distribution. The server provides service for customers’ in regular busy periods with parameter μ and under multiple working vacations, the server provides service with parameter μ_v with the exponential distribution. In this model, batches of customers are served as a group under the general bulk service rule, which was introduced by Neuts. In the batch service process, the service times for each customer within a batch may be independent and random variables. The number of customers’ in each batch can also vary. Thus, each batch of service contains a minimum of ’a’ units and a maximum of ’b’ units of customers’. Suppose that the number of  customers waiting in the queue is less than ’a’ server begins a vacation random variable V with parameter η, the breakdown β_v occurs during the busy state. This paper analyzed the steady-state equation, steady-state solutions, and measures of system performance. Specifically, various performance analyses, namely the mean length and other characteristics like the probability that the server is idle, regular busy and working vacation periods are analyzed. Finally, this paper computed the results with the working vacation and the classical multiple working vacation models.

References

Servi LD, Finn SG. M/M/1 Queue with Working Vacations (M/M/1/WV). Perform Evaluation. 2002; 50(1): 41-52.

Tian N, Wang K, Zhao X. The M/M/1 queue with single working vacation. Int J Inf Manag Sci. 2008; 19(4): 621-634. Available from:

Jain M, Jain A. Working Vacations Queuing Models with Multiple Types of Server Breakdowns. Appl Math. Model. 2010; 34(1): 1-13. https://doi.org/10.1016/j.apm.2009.03.019.

Choudhury G. A batch arrival queue with a vacation time under single vacation policy. Comput Oper Res. 2002; 29(14): 1941-1955. https://doi.org/10.1016/S0305-0548(01)00059-4.

Be Moussa MH, Abd Elmawgoud MTA, Elias ANE. Measuring Service Time Characteristics in Fast Food Restaurants in Cairo: A Case Study. Tour Today. 2015; 1(15): 90-104..

Abdelmawgoud MTA, Dawood AAA, Moussa MHB. The Impact of Prolonged Waiting Time of Food Service on Customers' Satisfaction. Minia J Tour Hosp Res. 2016; 1(1): 247-251. http://dx.doi.org/10.21608/mjthr.2016.262117.

Abid NA, Al-Madi AK. On The Queuing System M/Er/1/N. Baghdad Sci J. 2012; 9(2): 367–371. https://doi.org/10.21123/bsj.2012.9.2.367-371.

Berdjoudj L, Ameur L, Abbas K. Sensitivity analysis of the M/M/1 retrial queue with working vacations and vacation interruption. Int J Manag Sci Eng Manag. 2019; 14(4): 293-303. http://dx.doi.org/10.1080/17509653.2019.1566034.

Chakravarthy SR, Shruti, Kulshrestha R. A queueing model with server breakdowns, repairs, vacations, and backup server. Oper Res Perspect. 2020; 7: 1-13. https://doi.org/10.1016/j.orp.2019.100131.

Seenivasan M, Indumathi M, Chakravarthy VJ. Performance Analysis of Two Heterogeneous Server Queuing Model with Intermittently Obtainable Server Using Matrix Geometric Method. International Conference on Recent Trends in Applied Mathematical Sciences (ICRTAMS), 26-27 September 2020, Tiruvannamalai, India. J Phys.: Conf Ser 2020; 1724: 012001. https://doi.org/10.1088/1742-6596/1724/1/012001

Barbhuiya FP, Gupta UC. A Discrete-Time GIX/Geo/1 Queue with Multiple Working Vacations under Late and Early Arrival System. Methodol Comput Appl Probab. 2020; 22: 599-624. https://doi.org/10.1007/s11009-019-09724-6.

Kumar N, Gupta P. Cost Optimization of Single Server Retrial Queuing Model with Bernoulli Schedule Working Vacation, Vacation Interruption and Balking. J Math Comput Sci. 2021; 11(3): 2508-2523. https://doi.org/10.28919/jmcs/5552.

Gupta P, Kumar N. Performance Analysis of Retrial Queueing Model with Working Vacation, Interruption, Waiting Server, Breakdown and Repair. J Sci Res. 2021; 13(3): 833-844. https://doi.org/10.3329/jsr.v13i3.52546.

Agrawal P, Jain A, Madhu J. M/M/1 queuing model with working vacation and two type of server breakdown. 2nd National Conference on Recent Advancement in Physical Sciences, (NCRAPS) 2020 19-20 December 2020, Uttarakhand, INDIA. J Phys: Conf Ser. 2021; 1849: 012021. https://doi.org/10.1088/1742-6596/1849/1/012021 .

Seenivasan M, Abinaya. Markovian queuing model with single working vacation and catastrophic. Mater Today Proc. 2022; 51(8): 2348-2354. https://doi.org/10.1016/j.matpr.2021.11.572.

Ayyappan G, Meena S. Phase Type Queuing Model of Server Vacation, Repair and Degrading Service with Breakdown, Starting Failure and Close-Down. Reliab Theory Appl. 2023; 18(1(72)): 464-483. https://doi.org/10.24412/1932-2321-2023-172-464-483.

Agarwal R, Agarwal D, Upadhyaya S. Cost optimisation of a heterogeneous server queueing system with working breakdown using PSO. Int J Math Oper Res. 2023; 26(3): 410-424. https://doi.org/10.1504/IJMOR.2023.134842.

Somasundaram B, Karpagam S, Kumar KS, Kala R. Analysis of Priority Queueing System with Working Breakdown, Vacation and Vacation Interruption under Random Environment. Southeast Europe j. soft computing. 2023; 12(2): 57-66. http://dx.doi.org/10.21533/scjournal.v12i2.264.

Mathew N, Joshua VC, Krishnamoorthy A, Melikov A, Mathew AP. A production inventory model with server breakdown and customer impatience. Ann Oper Res. 2023; 331(2): 1269-1304. https://doi.org/10.1007/s10479-023-05659-x.

Arulmozhi N. (R2053) Analysis of MAP/PH/1 Queueing Model Subject to Two-stage Vacation Policy with Imperfect Service, Setup Time, Breakdown, Delay Time, Phase Type Repair and Reneging Customer. Appl Appl. Math Int J (AAM). 2023; 18(1): 1-33.

Ibraheem N, Hasan M. Combining Several Substitution Cipher Algorithms using Circular Queue Data Structure. Baghdad Sci J. 2020; 17(4): 1320. https://doi.org/10.21123/bsj.2020.17.4.1320.

Khan IE, Paramasivam R. Analysis of Batch Encouraged Arrival Markovian Model Due to a Secondary Optional Service, Break-Down and Numerous Vacations. Math Stat Eng Appl. 2023; 72(1): 1166-1177. https://doi.org/10.17762/msea.v72i1.2213.

Manoharan P, Raman KS. Impatient customers in a Markovian Queue with Multiple Working Vacation and Server Breakdown. J Pharm Negat Results. 2023; 14(2): 1729-1737. https://doi.org/10.47750/pnr.2023.14.02.218.

Su L, Wei J, Zhang X, Guo W, Zhang K. Traffic Breakdown Probability Estimation for Mixed Flow of Autonomous Vehicles and Human Driven Vehicles. Sens. 2023; 23(7): 1-14. https://doi.org/10.3390/s23073486.

Liu T-H, Hsu H-Y, Ke J-C, Chang F-M. Preemptive Priority Markovian Queue Subject to Server Breakdown with Imperfect Coverage and Working Vacation Interruption. Computation. 2023; 11(5): 89.page https://doi.org/10.3390/computation11050089.

Tian R, Wu X, He L, Han Y. Strategic Analysis of Retrial Queues with Setup Times, Breakdown and Repairs. Discrete Dyn Nat Soc. 2023: 2023: 13 pages. https://doi.org/10.1155/2023/4930414.

Xu X, Liu M, Zhao X. The bulk input M[X]/M/1 queue with working vacation. J Syst Sci Syst Eng. 2009; 18(3): 358-368. https://doi.org/10.1007/s11518-009-5111-4.

Lidiya P, Mary K. Performance study on heterogeneous arrival of batch service for multiple working vacations queuing system with breakdowns in the busy period. In 2023 First International Conference on Advances in Electrical, Electronics and Computational Intelligence (ICAEECI). IEEE Xplore. 2023; 1-6. https://doi.org/10.1109/ICAEECI58247.2023.10370986

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Analysing the Performance of  Queuing Model with the Busy Period Breakdown. Baghdad Sci.J [Internet]. [cited 2024 Dec. 6];22(2). Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/9155