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Well-posedness and asymptotic behavior of the time-dependent solution of an M/G/1 queueing model

  • Nurehemaiti Yiming
  • Geni Gupur
Article
  • 4 Downloads

Abstract

By using the Hille–Yosida theorem, Phillips theorem and Fattorini theorem we prove that the M/G/1 queueing model with vacations and multiple phases of operation, which is described by infinitely many partial differential equations with integral boundary conditions, has a unique positive time-dependent solution that satisfies the probability condition. Next, by studying the spectrum of the operator, which corresponds to the model, on the imaginary axis we prove that the time-dependent solution of the model strongly converges to its steady-state solution.

Keywords

M/G/1 queueing model with vacations and multiple phases of operation Time-dependent solution \(C_0\)-semigroup Resolvent set Eigenvalue 

Mathematics Subject Classification

Primary 47D03 47A10 Secondary 60K25 

References

  1. 1.
    Adams, R.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Doshi, B.: Queueing systems with vacations-a survey. Queueing Syst. 1, 29–66 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fattorini, H.O.: The Cauchy Problem. Addison-Wesley, Massachusetts (1983)zbMATHGoogle Scholar
  4. 4.
    Greiner, G.: Perturbing the bounding conditions of a generator. Houston J. Math. 13, 213–229 (1987)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gupur, G.: Well-posedness of M/G/1 queueing model with single vacations. Comput. Math. Appl. 44, 1041–1056 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gupur, G.: Advances in queueing models’ research. Acta Analysis Functionalis Applicata 13, 225–245 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gupur, G.: Point spectrum of the opreator corresponding to a reliability model and appication. J. Pseudo-Differ. Oper. Appl. 7, 411–429 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gupur, G., Li, X.Z., Zhu, G.T.: Functional Analysis Method in Queueing Theory. Research Information Ltd, Herdfortshire (2001)zbMATHGoogle Scholar
  9. 9.
    Gupur, G., Wong, M.W.: On a dynamical system for a reliability model. J. Pseudo-Differ. Oper. Appl. 2, 509–542 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Haji, A., Radl, A.: Asymptotic stability of the solution of the M/\(M^B\)/1 queueing model. Comput. Math. Appl. 53, 1411–1420 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kasim, E., Gupur, G.: Dynamic analysis of the M/G/1 queueing model with single working vacation. Int. J. Appl. Comput. Math. 3, 2803–2833 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li, J.J., Liu, L.W., Jiang, T.: Analysis of an M/G/1 queue with vacation and multiple phases of operation. Math. Methods Oper. Res. 87, 51–72 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Malogrosz, M.: Well-posedness and asymptotic behavior a multidimensional model of morphogen transport. J. Evol. Equ. 12, 353–366 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nagel, R.: One-Parameter Semigroups of Positive Operators (LNM 1184). Springer, Berlin (1986)Google Scholar
  15. 15.
    Pazoto, A.F., Souza, G.R.: On the well-posedness and asymptotic behavior of a nonlinear dispersive system in weighted spaces. Appl. Math. Optim. 69, 141–174 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Servi, L.D., Finn, S.G.: M/M/1 queue in with working vacations. Perform. Eval. 50, 41–52 (2002)CrossRefGoogle Scholar
  17. 17.
    Son, J., Yu, J.Y.: Population System Control. Springer, Berlin (1998)Google Scholar
  18. 18.
    Takagi, H.: Time-dependent analysis of M/G/l vacation models with exhaustive service. Queueing Syst. 6, 369–390 (1990)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Yechiali, U., Naor, P.: Queueing problems with hetergeneous arrivals and service. Oper. Res. 19, 722–734 (1971)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceXinjiang UniversityÜrümqiPeople’s Republic of China

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