This paper deals with the study of multi-server queueing model in a fuzzy environment with imposition of reneging of customers. Entry of the customers in the system is assumed to be Poisson process and exponential service time distribution under first - come-first-served basis. Specific of this investigation is to derive the various fuzzy performance measures such as fuzzy queue length, fuzzy waiting time in queue, fuzzy response time and fuzzy optimal number of servers in explicit form for the finite capacity multi-server queueing system by using recursive method. For the validity of the model we have obtained the numerical illustrations in tabular form which shows that fuzzy-queue can be more realistic than crisp queue.
KeywordsFuzzy EnvironmentPoissonOptimalReneging
Erlang, A.K. (1909) The Theory of Probabilities and Telephone Conversations. Nyt Tidsskrift for Matematik B, 20, 33-39.
Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Bellman, R.E. and Zadeh, L.A. (1970) Decision-Making in a Fuzzy Environment. Management Science, 17, B141-B164. https://doi.org/10.1287/mnsc.17.4.B141
Zadeh, L.A. (1978) Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems, 1, 3-28. https://doi.org/10.1016/0165-0114(78)90029-5
Prade, H.M. (1980) An Outline of Fuzzy or Possibilistic Models for Queueing Systems. In: Wang, P.P. and Chang, S.K., Eds., Fuzzy Sets, Springer, Boston, 147-153. https://doi.org/10.1007/978-1-4684-3848-2_13
Yager, R.R. (1986) A Characterization of the Extension Principle. Fuzzy Sets and Systems, 18, 205-217. https://doi.org/10.1016/0165-0114(86)90002-3
Li, R.J. and Lee, E.S. (1989) Analysis of Fuzzy Queues. Computers and Mathematics with Applications, 17, 1143-1147. https://doi.org/10.1016/0898-1221(89)90044-8
Buckley, J.J. (1990) Elementary Queueing Theory Based on Possibility Theory. Fuzzy Sets and Systems, 37, 43-52. https://doi.org/10.1016/0165-0114(90)90062-B
Negi, D.S. and Lee, E.S. (1992) Analysis and Simulation of Fuzzy Queues. Fuzzy Sets and Systems, 46, 321-330. https://doi.org/10.1016/0165-0114(92)90370-J
Kao, C., Li, C. and Chen, S. (1999) Parametric Programming to the Analysis of Fuzzy Queues. Fuzzy Sets and Systems, 107, 93-100. https://doi.org/10.1016/S0165-0114(97)00295-9
Buckley, J.J., Feuring, T. and Hayashi, Y. (2001) Fuzzy Queueing Theory Revisited. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 9, 527-537. https://doi.org/10.1142/S0218488501001046
Zimmermann, H.J. (2001) Fuzzy Set Theory and Its Applications. Springer Science and Business Media, Dordrecht. https://doi.org/10.1007/978-94-010-0646-0
Zhang, R., Phillis, Y.A. and Kouikoglou, V.S. (2005) Fuzzy Control of Queueing Systems. Springer-Verlag, London.
Dubois, D. and Prade, H. (2015) The Legacy of 50 Years of Fuzzy Sets: A Discussion. Fuzzy Sets and Systems, 281, 21-31. https://doi.org/10.1016/j.fss.2015.09.004
Chen, S.P. (2016) Time Value of Delays in Unreliable Production Systems with Mixed Uncertainties of Fuzziness and Randomness. European Journal of Operational Research, 25, 834-844. https://doi.org/10.1016/j.ejor.2016.06.021
Pardo, M.J. and Fuente, D. (2008) Design of a Fuzzy Finite Capacity Queueing Model Based on the Degree of Customer Satisfaction: Analysis and Fuzzy Optimization. Fuzzy Sets and Systems, 159, 3313-3332. https://doi.org/10.1016/j.fss.2008.05.019
Shahin, M., Doniavi, A., Solimanpura, M. and Shahin, M. (2015) A Novel Approach for Optimization in a Fuzzy Finite Capacity Queueing Model with System Cost and Expected Degree of Customer Satisfaction. Decision Science Letters, 4, 487-496. https://doi.org/10.5267/j.dsl.2015.6.001
Cruz, F.R.B. and Woensel, T.V. (2014) Finite Queueing Modeling and Optimization: A Selected Review. Journal of Applied Mathematics, 2014, Article ID: 374962. https://doi.org/10.1155/2014/374962
Fazlollahtabar, H. and Gholizadeh, H. (2019) Economic Analysis of the M/M/1/N Queueing System Cost Model in a Vague Environment. International Journal of Fuzzy Logic and Intelligent Systems, 9, 192-203. https://doi.org/10.5391/IJFIS.2019.19.3.192
Prameela, K.U. and Kumar, P. (2020) Conceptualization of Finite Capacity Single-Server Queueing Model with Triangular, Trapezoidal and Hexagonal Fuzzy Numbers using α-Cuts. In: Dutta, D. and Mahanty, B., Eds., Numerical Optimization in Engineering and Sciences, Vol. 979, Springer, Singapore, 201-212. https://doi.org/10.1007/978-981-15-3215-3_19
Lin, C.H. and Ke, J.C. (2009) Optimal Operating Policy for a Controllable Queueing Model with a Fuzzy Environment. Journal of Zhejiang University-Science A, 10, 311-318. https://doi.org/10.1631/jzus.A0820139
Pardo, M.J. and Fuente, D. (2009) A New Technique to Optimize the Functions of Fuzzy Profit of Queueing Models: Application to a Queueing Model with Publicity and Renouncement. Computers and Mathematics with Applications, 57, 850-864. https://doi.org/10.1016/j.camwa.2008.10.091
Azadeh, A., Ebrahim, R.M. and Eivazy, H. (2010) Parameter Optimization of Tandem Queue Systems with Finite Intermediate Buffers via Fuzzy Simulation. Performance Evaluation, 67, 353-360. https://doi.org/10.1016/j.peva.2009.10.004
Zhao, F., Li, Y.P. and Huang, G.H. (2013) A Queue-Based Interval-Fuzzy Programming Approach for Electric-Power Systems Planning. International Journal of Electrical Power and Energy Systems, 47, 337-350. https://doi.org/10.1016/j.ijepes.2012.11.006
Gonzalez-Lopez, V.A., Gholizadeh, R. and Shirazi, A.M. (2016) Optimization of Queueing Theory Based on Vague Environment. International Journal of Fuzzy System Applications, 5, 1-26. https://doi.org/10.4018/IJFSA.2016010101
De, S.K. and Mahata, G.C. (2019) A Cloudy Fuzzy Economic Order Quantity Model for Imperfect-Quality Items with Allowable Proportionate Discounts. Journal of Industrial Engineering International, 15, 571-583. https://doi.org/10.1007/s40092-019-0310-1
Gholizadeh, H., Tajdin, A. and Javadian, N. (2020) A Closed-loop Supply Chain Robust Optimization for Disposable Appliances. Neural Computing and Applications, 32, 3967-3985. https://doi.org/10.1007/s00521-018-3847-9
Nayeri, S., Tavakoli, M., Tanhaeean, M. and Jolai, F. (2021) A Robust Fuzzy Stochastic Model for the Responsive-Resilient Inventory-Location Problem: Comparision of Metaheuristic Algorithms. Annals of Operations Research, 2021, 1-41. https://doi.org/10.1007/s10479-021-03977-6
Haight, F.A. (1957) Queueing with Balking. Biometrika, 44, 360-369. https://doi.org/10.1093/biomet/44.3-4.360
Haight, F.A. (1959) Queueing with Reneging. Metrika, 2, 186-197. https://doi.org/10.1007/BF02613734
Abou-El-Ata, M.O. and Hariri, A.M.A. (1992) The M/M/c/N Queue with Balking and Reneging. Computers and Operations Research, 19, 713-716. https://doi.org/10.1016/0305-0548(92)90010-3
Wang, K., Li, N. and Jiang, Z. (2010) Queueing System with Impatient Customers: A Review. Proceedings of 2010 IEEE International Conference on Service Operations and Logistics, and Informatics, QingDao, 15-17 July 2010, 82-87. https://doi.org/10.1109/SOLI.2010.5551611
Bouchentouf, A.A., Cherfaoui, M. and Boualem, M. (2019) Performance and Economic Analysis of a Single Server Feedback Queueing Model with Vacation and Impatient Customers. OPSEARCH, 56, 300-323. https://doi.org/10.1007/s12597-019-00357-4
Bhardwaj, R., Singh, T.P. and Kumar, V. (2018) Mathematical Study of Queue System with Impatient Customers under Fuzzy Environment. In: Pant, M., Ray, K., Sharma, T., Rawat, S. and Bandyopadhyay, A., Eds., Soft Computing: Theories and Applications, Vol. 583, Springer, Singapore, 679-688. https://doi.org/10.1007/978-981-10-5687-1_60
Chen, G., Liu, Z. and Zhang, J. (2020) Analysis of Strategic Customer Behavior in Fuzzy Queueing Systems. Journal of Industrial and Management Optimization, 16, 371-386. https://doi.org/10.3934/jimo.2018157