International Journal of Nonlinear Analysis and Applications

Developing bulk arrival queuing models with the constant batch policy under uncertainty data using (0 - 1) variables

Document Type : Research Paper

Author

Department of Studies and Planning Department, University of Baghdad, Baghdad, Iraq

Abstract

This paper delves into some significant performance measures (PMs) of a bulk arrival queueing system with constant batch size b, according to arrival rates and service rates being fuzzy parameters. The bulk arrival queuing system deals with observation arrival into the queuing system as a constant group size before allowing individual customers entering to the service. This leads to obtaining a new tool with the aid of generating function methods. The corresponding traditional bulk queueing system model is more convenient under an uncertain environment. The $\alpha$-cut approach is applied with the conventional Zadeh's extension principle (ZEP) to transform the triangular membership functions (Mem. Fs) fuzzy queues into a family of conventional bulk
queues. This new model focus on mixed-integer non-linear programming (MINLP) tenders a mathematical computational approach is known as (0 -1) variables. To measures the efficiency of the method, the efficient solution strategy plays a crucial role in the adequate application of these techniques. Furthermore, different stages of the $\alpha$-cut intervals were analyzed and the final part of the article gives a numerical solution of the proposed model to achieve practical issues.

Keywords

[1] L. Abolnikov and J.H. Dshalalow, On a multilevel controlled bulk queuing system MX/Gr, R/11, J. Appl. Math. Stoch. Anal. 3 (1992) 237–260.
[2] K.R. Balachandran, Control policies for a single server system, Manag. Sci. 9 (1973) 1013–1018.[3] J.J. Buckley, Elementary queuing theory based on possibility theory, Fuzzy Set Syst. 37 (1990) 43–52.
[4] M.L. Chaudhry and J.G.C. Templeton, A First Course in Bulk Queues, John Wiley & Sons, New York, 1983.
[5] S.P. Chen, Parametric Nonlinear Programming Approach to Fuzzy Queues with Bulk Service, European J. Operat. Res. 163 (2005) 434–444.
[6] S. Chen, A bulk arrival queuing model with fuzzy parameters and varying batch sizes, Appl. Math. Model. 9 (2006) 920–929.
[7] S.P Chen, Measuring performances of multiple-channel queuing systems with imprecise data: a membership function approach, J. Operat. Res. Soc.,59 (2008) 381-387.
[8] S. Ghimire, R.P. Ghimire, and G.B. Thapa, Mathematical models of Mb/M/1 bulk arrival queuing system, J. Instit. Engin. 10 (2014) 184–191.
[9] G. Choudhury, A batch arrival queue with a vacation time under a single vacation policy, Comput. Operat. Res. 29 (2002) 1941–1955.
[10] G. Choudhury, An MX/G/1 queuing system with a setup period and a vacation period, Queuing Syst. 36 (2000) 23–38.
[11] D. Gross and C.M. Harris, Fundamentals of Queuing Theory, Third Ed., Wiley, New York, 1998.
[12] I.E. Grossmann, Mixed-integer nonlinear programming techniques for the synthesis of engineering systems, Res. Engin. Design, 1 (1990) 205–228.
[13] S. M. Gupta, Interrelationship between Controlling Arrival, and Service in Queuing Systems, Comput. Operat. Res. 22 (1995) 1005–1014.
[14] C. Kao, C. Li and S. Chen, Parametric programming to the analysis of fuzzy queues, Fuzzy Sets Syst. 107 (1999) 93–100.
[15] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Academic Press, New York, 1975.
[16] J.C. Ke, Bi-level control for batch arrival queues with an early setup and unreliable server, Appl. Math. Model 28 (2004) 469–485.
[17] J.C. Ke, F.M. Chang, and C.J. Chang controlling arrivals for a Markovian queuing system with a second optional service, Int. J. Indust. Engin. Theory Appl. Pract. 17 (2010) 48–57.
[18] R.F. Khalaf, K.C. Madan, and C.A. Lukas, An M[x]/G/1 queue with Bernoulli schedule general vacation times, general extended vacations, random breakdowns, general delay times for repairs to start and general repair times, J. Math. Res. 3 (2011) 8–20.
[19] V.A. Kumar, A membership function solution approach to fuzzy queue with Erlang service model, Int. J. Math. Sci. Appl. 1 (2011).
[20] C.H. Lin and J.C. Ke, Optimal Operating Policy for a Controllable Queuing Model With a Fuzzy Environment, J. Zhejiang University Sci., 10 (2009) 311-318.
[21] R.J. Li and E.S. Lee, Analysis of fuzzy queues, Comput. Math. Appl. 17 (1989) 1143–1147.
[22] M.K. Mary, J. Rose and T. Gokilavani, Analysis of Mx/M/1/MWV with fuzzy parameters, Int. J. Comput. Appl. 4 (2014).
[23] Z. Mueen, R. Ramli and N. Zura, Parametric nonlinear programming approach with fuzzy queues using hexagonal membership functions, J. Comput. Theor. Nanosci. 14 (2017) 4979-4985.
[24] H. M. Park, T.S. Kim and K.C. Cha, Analysis of a two-phase queuing system with a constant-size batch policy, European J. Operat. Res. 1 (2010) 118–122.
[25] S. Wang and D. Wang, A membership function solution to multiple-server fuzzy queues, Proc. Int. Conf. Serv. Syst. Serv. Manag. 1 (2005).
[26] R.R. Yager, A characterization of the extension principle, Fuzzy Sets Syst. 18 (1986) 205–217.
[27] D. Yang and P. Chang, A parametric programming solution to the F-policy queue with fuzzy parameters, Int. J. Syst. Sci. 46 (2015) 590-598.
[28] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst. 1 (1978) 3–28.
[29] H. J. Zimmermann, Fuzzy Set Theory and Its Applications, Fourth Ed., Kluwer Academic, Boston 2001.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 1113-1121
  • Receive Date: 02 March 2021
  • Revise Date: 14 April 2021
  • Accept Date: 26 May 2021