Analysis of the M/G/1 queueing model with second optional service and server breakdowns


We investigate an M/G/1 queue with second optional service and server breakdowns, which is described by infinitely many partial differential equations with boundary conditions and initial conditions. Firstly, by using \(C_0\)-semigroup theory we prove well-posedness of the system and the system has a unique positive time-dependent solution that satisfies the probability condition. Next, by studying spectral properties of the operator corresponding to the model we prove that all points on the imaginary axis except zero belong to the resolvent set of the operator. Lastly, we prove that zero is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model strongly converges to zero.

This is a preview of subscription content, log in to check access.


  1. 1.

    Wang, J.T.: An M/G/1 queue with second optional service and server breakdowns. Comput. Math. Appl. 47, 1713–1723 (2004)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Cao, J., Cheng, K.: Analysis of M/G/1 queueing system with repairable service station. Acta Math. Appl. Sin. 5, 113–127 (1982)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Li, W., Shi, D., Chao, X.: Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations. J. Appl. Probab. 34, 546–555 (1997)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Wang, J., Cao, J., Li, Q.: Reliability analysis of the retrial queue with server breakdowns and repairs. Queueing Syst. 38, 363–380 (2001)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Doshi, B.: Queueing systems with vacations—a survey. Queueing Syst. 1, 29–66 (1986)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Madan, K.C.: An M/G/1 queue with second optional service. Queueing Syst. 34, 37–46 (2000)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Gupur, G.: Analysis of the M/G/1 retrial queueing model with server breakdowns. J. Pseudo-Differ. Oper. Appl. 3, 313–340 (2010)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Gupur, G., Kasim, E.: Time-dependent analysis for the \(M^{[X]}/G/1\) retrial queueing model with server breakdowns and constant rate of repeated attempts. Int. J. Pure Appl. Math. 63, 419–441 (2010)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Gupur, G., Ehmet, R.: Asymptotic behavior of the time-dependent solution of an M/G/1 queueing model. Bound. Value Probl. 2013, Article ID 17 (2013).

  10. 10.

    Haji, A., Radl, A.: A semigroup approach to queueing systems. Semigroup Forum 75, 610–624 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Haji, A., Radl, A.: Asymptotic stability of the solution of the \(M/M^B/1\) queueing model. Comput. Math. Appl. 53, 1411–1420 (2007)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Haji, A., Radl, A.: A semigroup approach to the Gnedenko system with single vacation of a repairman. Semigroup Forum 86, 41–57 (2013)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Adams, R.A.: Sobolev Space. Academic Press, New York (1975)

    Google Scholar 

  14. 14.

    Greiner, G.: Perturbing the boundary conditions of a generator. Houston J. Math. 13, 213–229 (1987)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)

    Google Scholar 

  16. 16.

    Nagel, R.: One-Parameter Semigroups of Positive Operators. Springer, Heidelberg (1986)

    Google Scholar 

  17. 17.

    Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306, 837–852 (1988)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Abdukerim Haji.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the National Natural Science Foundation of China (Nos: 11361057, 11761066).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hoshur, A., Haji, A. Analysis of the M/G/1 queueing model with second optional service and server breakdowns. J. Pseudo-Differ. Oper. Appl. 11, 1265–1287 (2020).

Download citation


  • M/G/1 queueing model with second optional service and server breakdowns
  • \(C_0\)-semigroup
  • Time-dependent solution
  • Dispersive operator
  • Eigenvalue

Mathematics Subject Classification

  • 47D06
  • 90B25