Advertisement

Journal of Zhejiang University-SCIENCE A

, Volume 10, Issue 2, pp 311–318 | Cite as

Optimal operating policy for a controllable queueing model with a fuzzy environment

  • Chuen-horng Lin
  • Jau-chuan Ke
Article

Abstract

We construct the membership functions of the fuzzy objective values of a controllable queueing model, in which cost elements, arrival rate and service rate are all fuzzy numbers. Based on Zadeh’s extension principle, a set of parametric nonlinear programs is developed to find the upper and lower bounds of the minimal average total cost per unit time at the possibility level. The membership functions of the minimal average total cost are further constructed using different values of the possibility level. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the object value is expressed and governed by the membership functions, the optimization problem in a fuzzy environment for the controllable queueing models is represented more accurately and analytical results are more useful for system designers and practitioners.

Key words

Controllable queue Fuzzy sets Membership function Nonlinear programming 

CLC number

O22 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arumuganathan, R., Jeyakumar, S., 2005. Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times. Appl. Math. Model., 29(10):972–986. [doi:10.1016/j.apm.2005.02.013]CrossRefMATHGoogle Scholar
  2. Bell, C.E., 1971. Characterization and computation of optimal policies for operating an M/G/1 queueing system with removable server. Oper. Res., 19(1):208–218. [doi:10.1287/opre.19.1.208]CrossRefMATHGoogle Scholar
  3. Buckley, J.J., Feuring, T., Hayashi, Y., 2001. Fuzzy queueing theory revisited. Int. J. Uncert., Fuzz., Knowl.-Based Syst., 9(5):527–537. [doi:10.1142/S0218488501001046]MathSciNetCrossRefMATHGoogle Scholar
  4. Buzacott, J., Shanthikumar, J., 1993. Stochastic Models of Manufacturing Systems. Prentice-Hall, Englewood Cliffs, NJ.MATHGoogle Scholar
  5. Choudhury, G., Madan, K.C., 2005. A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy. Math. Comput. Model., 42(1–2):71–85. [doi:10.1016/j.mcm.2005.04.003]MathSciNetCrossRefMATHGoogle Scholar
  6. Choudhury, G., Paul, M., 2006. A batch arrival queue with a second optional service channel under N-policy. Stoch. Anal. Appl., 24(1):1–21. [doi:10.1080/07362990500397277]MathSciNetCrossRefMATHGoogle Scholar
  7. Gal, T., 1979. Postoptimal Analysis, Parametric Programming, and Related Topics. McGraw-Hill, New York.MATHGoogle Scholar
  8. Gross, D., Harris, C.M., 1998. Fundamentals of Queueing Theory (3rd Ed.). John Wiley, New York.MATHGoogle Scholar
  9. Heyman, D.P., 1968. Optimal operating policies for M/G/1 queueing system. Oper. Res., 16(2):362–382. [doi:10.1287/opre.16.2.362]CrossRefMATHGoogle Scholar
  10. Hillier, F.S., Lieberman, G.J., 2001. Introduction to Operations Research (7th Ed.). McGraw-Hill, Singapore.MATHGoogle Scholar
  11. Kaufmann, A., 1975. Introduction to the Theory of Fuzzy Subsets, Volume 1. Academic Press, New York.MATHGoogle Scholar
  12. Kella, O., 1989. The threshold policy in the M/G/1 queue with server vacations. Nav. Res. Logist., 36(1):111–123. [doi:10.1002/1520-6750(198902)36:1〈111::AID-NAV3220360109〉3.0.CO;2-3]MathSciNetCrossRefMATHGoogle Scholar
  13. Kleinrock, L., 1975. Queueing Systems, Vol. 1: Theory. Wiley, New York.MATHGoogle Scholar
  14. Lee, H.S., Srinivasan, M.M., 1989. Control policies for the M [x]/G/1 queueing system. Manag. Sci., 35(6):708–721. [doi:10.1287/mnsc.35.6.708]MathSciNetCrossRefMATHGoogle Scholar
  15. Lee, H.W., Park, J.O., 1997. Optimal strategy in N-policy production system with early set-up. J. Oper. Res. Soc., 48(3):306–313. [doi:10.1057/palgrave.jors.2600354]CrossRefMATHGoogle Scholar
  16. Lee, H.W., Lee, S.S., Chae, K.C., 1994a. Operating characteristics of M X/G/1 queue with N policy. Queueing Systems, 15(1–4):387–399. [doi:10.1007/BF01189247]MathSciNetCrossRefMATHGoogle Scholar
  17. Lee, H.W., Lee, S.S., Park, J.O., Chae, K.C., 1994b. Analysis of M [x]/G/1 queue with N policy and multiple vacations. J. Appl. Probab., 31(2):467–496. [doi:10.2307/3215040]MathSciNetMATHGoogle Scholar
  18. Lee, S.S., Lee, H.W., Yoon, S.H., Chae, K.C., 1995. Batch arrival queue with N-policy and single vacation. Comput. Oper. Res., 22(2):173–189. [doi:10.1016/0305-0548(94)E0015-Y]CrossRefMATHGoogle Scholar
  19. Li, R.J., Lee, E.S., 1989. Analysis of fuzzy queues. Comput. Math. Appl., 17(7):1143–1147. [doi:10.1016/0898-1221(89)90044-8]MathSciNetCrossRefMATHGoogle Scholar
  20. Pearn, W.L., Ke, J.C., Chang, Y.C., 2004. Sensitivity analysis of the optimal management policy for a queueing system with a removable and non-reliable server. Comput. Ind. Eng., 46(1):87–99. [doi:10.1016/j.cie.2003.11.001]CrossRefGoogle Scholar
  21. Tadj, L., Choudhury, G., 2005. Optimal design and control of queues. TOP, 13(2):359–414. [doi:10.1007/BF02579061]MathSciNetCrossRefMATHGoogle Scholar
  22. Tadj, L., Choudhury, G., Tadj, C., 2006a. A quorum queueing system with a random setup time under N-policy with Bernoulli vacation schedule. Qual. Technol. Quantit. Manag., 3(2):145–160.MathSciNetCrossRefMATHGoogle Scholar
  23. Tadj, L., Choudhury, G., Tadj, C., 2006b. A bulk quorum queueing system with a random setup time under N-policy with Bernoulli vacation schedule. Stoch.: Int. J. Probab. Stoch. Processes, 78(1):1–11. [doi:10.1080/17442500500397574]MathSciNetMATHGoogle Scholar
  24. Taha, H.A., 2003. Operations Research: An Introduction (7th Ed.). Prentice-Hall, New Jersey.MATHGoogle Scholar
  25. Wang, K.H., Kao, H.T., Chen, G., 2004. Optimal management of a removable and non-reliable server in an infinite and a finite M/H k/1 queueing system. Int. J. Qual. Technol. Quantit. Manag., 1(2):325–339.MathSciNetCrossRefGoogle Scholar
  26. Yadin, M., Naor, P., 1963. Queueing systems with a removable service station. J. Oper. Res. Soc., 14(4):393–405. [doi:10.1057/jors.1963.63]CrossRefGoogle Scholar
  27. Yager, R.R., 1986. A characterization of the extension principle. Fuzzy Sets Syst., 18(3):205–217. [doi:10.1016/0165-0114(86)90002-3]MathSciNetCrossRefMATHGoogle Scholar
  28. Zadeh, L.A., 1978. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst., 1(1):3–28. [doi:10.1016/0165-0114(78)90029-5]MathSciNetCrossRefMATHGoogle Scholar
  29. Zhang, B., 2006. A Fuzzy-logic-based Methodology for Batch Process Scheduling. IEEE Systems and Information Engineering Design Symp., p.101–105. [doi:10.1109/SIEDS.2006.278721]Google Scholar
  30. Zhang, R., Phillis, Y.A., Kouikoglou, V.S., 2005. Fuzzy Control of Queueing Systems. Springer, New York.MATHGoogle Scholar
  31. Zimmermann, H.J., 2001. Fuzzy Set Theory and Its Applications (4th Ed.). Kluwer Academic, Boston.CrossRefGoogle Scholar

Copyright information

© Zhejiang University and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Department of Information ManagementNational Taichung Institute of TechnologyTaichungTaiwanChina
  2. 2.Department of Applied StatisticsNational Taichung Institute of TechnologyTaichungTaiwanChina

Personalised recommendations