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A STEADY-STATE BEHAVIOUR OF AN $M/M/1$ QUEUE WITH OPTIONAL DIFFERENTIATED WORKING VACATIONS, SERVER BREAKDOWN, AND CUSTOMER BALKING
Keywords:
single server queue, optional working vacations, breakdown, balking, stability analysis and optimizationDOI:
https://doi.org/10.17654/0972361725025Abstract
This paper analyses the steady-state behaviour of an $M/M/1$ queueing system incorporating optional differentiated working vacations, server breakdowns, and customer balking. Following a busy period, the server might transition into an optional I or II working vacation, leading to a reduction in the service rate. Customers may balk when the server is on vacation or experiencing a breakdown. The model’s stable solution is derived using probability-generating functions, enabling the evaluation of key performance metrics. A cost optimization framework is employed to minimize the cost by balancing system parameters. Numerical examples demonstrate the impact of varying arrival and service rates on customer queue length and waiting times.
Received: October 1, 2024
Revised: December 16, 2024
Accepted: January 7, 2025
References
P. K. Agrawal, A. Jain and M. Jain, queueing model with working vacation and two type of server breakdown, Journal of Physics: Conference Series, 1849(1) 2021, pp. 012021. DOI: 10.1088/1742-6596/1849/1/012021.
G. Ayyappan and S. Meena, Phase type queueing model of server vacation, repair and degrading service with breakdown, starting failure and close-down, Reliability: Theory and Applications 18(172) (2023), 464-483.
DOI: 10.24412/1932-2321-2023-172-464-483.
V. M. Chandrasekaran, K. Indhira, M. C. Saravanarajan and P. Rajadurai, A survey on working vacation queueing models, International Journal of Pure and Applied Mathematics 106(6) (2016), 33-41. DOI: 10.12732/ijpam.v106i6.5.
B. T. Doshi, Queueing systems with vacations -A survey, Queueing Systems 1(1) (1986), 29-66. DOI: https://doi.org/10.1007/BF01149327.
C. Goswami and N. Selvaraju, A working vacation queue with priority customers and vacation interruptions, International Journal of Operational Research 17(3) (2013), 311-332. DOI: https://doi.org/10.1504/IJOR.2013.054438.
P. Gupta and N. Kumar, Performance analysis of retrial queueing model with working vacation, interruption, waiting server, breakdown and repair, J. Sci. Res. 13(3) (2021), 833-844. DOI: http://dx.doi.org/10.3329/jsr.v13i3.52546.
O. C. Ibe and O. A. Isijola, multiple vacation queueing systems with differentiated vacations, Modelling and Simulation in Engineering 1 (2014), 158247. DOI: https://doi.org/10.1155/2014/158247.
M. Jain and A. Jain, Working vacations queueing model with multiple types of server breakdowns, Applied Mathematical Modelling 34(1) (2010), 1-13.
DOI: https://doi.org/10.1016/j.apm.2009.03.019.
J. Ebenesar Anna Bagyam, P. Suganthi and P. Visali, Unreliable single server retrial queueing model with repeated vacation, International Journal of Mathematics in Operational Research 26(3) (2023), 357-372.
DOI: https://doi.org/10.1504/IJMOR.2023.134838.
M. Jain and Preeti, Cost analysis of a machine repair problem with standby, working vacation and server breakdown, International Journal of Mathematics in Operational Research 6(4) (2014), 437-451.
DOI: https://doi.org/10.1504/IJMOR.2014.063156.
V. Karthick and V. Suvitha, Time dependent behaviour of a single server queueing system with differentiated working vacations subject to system disaster, Reliability: Theory and Applications 18(4)(76) (2023), 1019-1031.
V. Karthick, V. Suvitha and R. Kalyanaraman, A single server Markovian differentiated working vacation queue with server breakdown, International Journal of Mathematics in Operational Research 27(1) (2024), 105-120.
DOI: https://doi.org/10.1504/IJMOR.2024.136862.
J. C. Ke, The optimal control of an queueing system with server vacations, startup and breakdowns, Computers and Industrial Engineering 44(4) (2003), 567-579. DOI: https://doi.org/10.1016/S0360-8352(02)00235-8.
J. C. Ke and W. L. Pearn, Optimal management policy for heterogeneous arrival queueing systems with server breakdowns and vacations, Quality Technology and Quantitative Management 1(1) (2004), 149-162.
DOI: https://doi.org/10.1080/16843703.2004.11673069.
P. V. Laxmi, V. Goswami and K. Jyothsna, Analysis of finite buffer Markovian queue with balking, reneging and working vacations, International Journal of Strategic Decision Sciences (IJSDS) 4(1) (2013), 1-24.
DOI: https://doi.org/10.4018/jsds.2013010101.
P. V. Laxmi and K. Jyothsna, Performance analysis of variant working vacation queue with balking and reneging, International Journal of Mathematics in Operational Research 6(4) (2014), 505-521.
DOI: https://doi.org/10.1504/IJMOR.2014.063158.
P. Rajadurai, M. C. Saravanarajan and V. M. Chandrasekaran, Cost optimization analysis of retrial queue with k optional phases of service under multiple working vacations and random breakdowns, International Journal of Industrial and Systems Engineering 29(2) (2018), 193-222.
DOI: https://doi.org/10.1504/IJISE.2018.091900.
A. A. Rao, V. R. Devi and K. Chandan, Optimal strategy analysis of N-policy two-phase vacation queueing system with server start-up, time-out and breakdown, Journal of Physics: Conference Series, 1344(1) (2019), pp. 012025.
DOI: 10.1088/1742-6596/1344/1/012025.
M. Seenivasan and S. Chandiraleka, Single server queueing model with multiple working vacation and breakdown, 2022 Second International Conference on Advances in Electrical, Computing, Communication and Sustainable Technologies (ICAECT), IEEE, 2022, pp. 1-5. DOI: 10.1109/ICAECT54875.2022.9807852.
L. D. Servi and S. G. Finn, queues with working vacations, Performance Evaluation 50(1) (2002), 41-52.
DOI: https://doi.org/10.1016/S0166-5316(02)00057-3.
M. I. G. Suranga Sampath and J. Liu, Impact of customers impatience on an queueing system subject to differentiated vacations with a waiting server, Quality Technology and Quantitative Management 17(2) (2018), 125-148.
DOI: https://doi.org/10.1080/16843703.2018.1555877.
T. Y. Wang, J. C. Ke and F. M. Chang, Analysis of a discrete-time queue with server subject to vacations and breakdowns, Journal of Industrial and Production Engineering 30(1) (2013), 54-66.
DOI: https://doi.org/10.1080/10170669.2012.762060.
D. Y. Yang and C. H. Wu, Cost-minimization analysis of a working vacation queue with n-policy and server breakdowns, Computers and Industrial Engineering 82 (2015), 151-158. DOI: https://doi.org/10.1016/j.cie.2015.01.017.
K. V. Vijayashree and K. Ambika, An queueing model subject to differentiated working vacation and customer impatience, Computational Intelligence, Cyper Security and Computational Models, Models and Techniques for Intelligent Systems and Automation, 4th International Conference, ICC3 2019, Coimbatore, India, December 19-21, 2019, Revised Selected Papers 4, 107-122. Springer, Singapore, 2020. DOI: https://doi.org/10.1007/978-981-15-9700-8_9.
H. Zhang and G. Zhou, queue with m kinds of differentiated working vacations, Journal of Applied Mathematics and Computing 54 (2017), 213-227. DOI: https://doi.org/10.1007/s12190-016-1005-z.
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