Analysis of M/M/1 Model with Complete Breakdown during Busy Period, Optional Server with Limited Service, Complete Vacation and Delay in Repair
R K Shrivastava
Shrivastava
Rachna Rathore
Research Scholar, Shrimant Madhav Rao Scindia Govt. Model Science College, Jiwaji University, Gwalior, M.P. India.
13-22
Vol: 14, Issue: 3, 2024
Receiving Date:
2024-04-30
Acceptance Date:
2024-08-04
Publication Date:
2024-08-13
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http://doi.org/10.37648/ijrst.v14i03.002
Abstract
In this paper we have analyzed a single server Markovian queueing model with an optional server for limited-service time, complete breakdown during busy period, complete vacation with some delay in repair. Customers arrival follow Poisson’s distribution with rate λ. Service time during busy period is exponentially distributed with rate μ. The server goes under complete breakdown during busy period and hence sent for repairing. During breakdown an optional server with limited-service time is available for serving customers rather than stopping service. This optional server after completing busy period moves to working vacation for limited time period, where service time during this period is slower than busy period service time as server has some prior commitments or work to finish. As the limited-service time finishes, the server goes for a complete vacation and will not serve any customer during this period. If main server gets repaired, it immediately resumes busy period but if some delay occurs in repairing of main server, then optional server (after completing vacation) will act as main server and resumes busy period until main server get repaired. The closed form expression of various system probabilities is derived. Various system performance measures like waiting time, queue length have been evaluated. Finally, some numerical and graphical results have been shown to model the impact of some parameters on different performance measures.
Keywords:
Busy Period; Complete Breakdown; Complete Vacation; Delayed Repair; Limited Service; Optional Server
References
- Gabi Hanukov and Shraga Shoval(2023), A Model for a Vacation Queuing Policy considering Server’s Deterioration and Recovery, Mathematics, 11(12), 2640; https://doi.org/10.3390/math11122640.
- GnanaSekar MMN, Kandaiyan I (2022), Analysis of an M/G/1 Retrial Queue with Delayed Repair and Feedback under Working Vacation policy with Impatient Customers. Symmetry, 2022;14(10):2024, https://doi.org/10.3390/sym14102024.
- M. Seenivasan, H.Manikandan, J.S. Epciya(2022) , M/M/1 Queue with Server Breakdown, Single Working Vacation, State Reliant Customers and Feedback, Second International Conference on Advances in Electrical, Computing, Communication and Sustainable Technologies (ICAECT).
- Roma Rani Dasa, V.N. Rama Devi, Abhishek Rathore, K. Chandan (2022), Analysis of Markovian Queueing System with Server Failures, N-Policy and Second Optional Service. International Journal of Nonlinear Analysis and Applications, 13 (2022), No. 1, 3073-3083,
- Agassi Melikov, Sevinj Aliyeva and Janos Sztrik (2021), Retrial Queues with Unreliable Servers and Delayed Feedback, Mathematics 2021, 9(19), 2415, Page no.1-23.
- Poonam Gupta (2021), Study of Feedback Retrial Queueing System with Working Vacation, Setup Time and Perfect Repair, Ratio Mathematica Volume 41, 2021, pp.291-307.
- P. Gupta, N. Kumar (2021), Performance analysis of Retrial Queueing Model with Working Vacation, Interruption, Waiting Server, Breakdown and Repair, Journal of Scientific Research, 13 (3), 833-844 (2021).
- Chakravarthy, Srinivas R.; Shruti; Kulshrestha, Rakhee (2020), A Queueing Model with Server Breakdowns, Repairs, Vacations, and Backup Server, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 7, pp. 1-13, 100131.
- Bharathidass S, Arivukkarasu V and Ganesan V (2018), Bulk Service Queue with Server Breakdown and Repairs, International Journal of Statistics and Applied Mathematics 2018; 3(1): 136-142.
- D.H. Yang, and D. Wu (2015), Cost-minimization Analysis of a Working Vacation Queue with N-policy and Server Breakdowns, Computers & Industrial Engineering, 82, pp. 151-158.
- Begum, Afthab et al. (2014) ,N Policy for Repairable Bulk Arrival Queueing Model with Setup, Second Multi Optional Service Facility under Restricted Number of Server Vacations, International Journal of Innovative Research in Science, Engineering and Technology 3 (2014): pp. 14614-14626.
- N. Tian and Z. G. Zhang (2006), Vacation Queueing Models: Theory and Applications, Springer.
- L.D. Servi and S.G. Finn (2002), M/M/1 Queue with Working Vacations (M/M/1/WV), Performance Evaluation, vol 50, pp 41–52.
- Awi Federgruen and Kut C. So (1991), Optimality of Threshold Policies in Single-Server Queueing Systems with Server Vacations, Advances in Applied Probability, Vol. 23, No. 2, pp. 388-405.
- B. T. Doshi. (1990) ,Single server queues with vacations, Stochastic analysis of Computer and Communication Systems, pages 217–265, 1990.
- B. T. Doshi. (1986), Queueing systems with vacations- a survey, Queueing systems: Theory and Applications, volume 1, pages 29–66, 1986.
- Marcel F. Neuts, David M Lucantoni (1979), A Markovian Queue with N Servers subject to Breakdowns and Repairs, Management Science,25(9):849-861.
- Y. Levy and U. Yechiali (1975), Utilization of Idle Time in an M/G/1 Queueing System, Management Science, vol 22, pp 202–211.
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