Abstract
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.
Similar content being viewed by others
References
Wang, J.T.: An M/G/1 queue with second optional service and server breakdowns. Comput. Math. Appl. 47, 1713–1723 (2004)
Cao, J., Cheng, K.: Analysis of M/G/1 queueing system with repairable service station. Acta Math. Appl. Sin. 5, 113–127 (1982)
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)
Wang, J., Cao, J., Li, Q.: Reliability analysis of the retrial queue with server breakdowns and repairs. Queueing Syst. 38, 363–380 (2001)
Doshi, B.: Queueing systems with vacations—a survey. Queueing Syst. 1, 29–66 (1986)
Madan, K.C.: An M/G/1 queue with second optional service. Queueing Syst. 34, 37–46 (2000)
Gupur, G.: Analysis of the M/G/1 retrial queueing model with server breakdowns. J. Pseudo-Differ. Oper. Appl. 3, 313–340 (2010)
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)
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). https://doi.org/10.1186/1687-2770-2013-17
Haji, A., Radl, A.: A semigroup approach to queueing systems. Semigroup Forum 75, 610–624 (2007)
Haji, A., Radl, A.: Asymptotic stability of the solution of the \(M/M^B/1\) queueing model. Comput. Math. Appl. 53, 1411–1420 (2007)
Haji, A., Radl, A.: A semigroup approach to the Gnedenko system with single vacation of a repairman. Semigroup Forum 86, 41–57 (2013)
Adams, R.A.: Sobolev Space. Academic Press, New York (1975)
Greiner, G.: Perturbing the boundary conditions of a generator. Houston J. Math. 13, 213–229 (1987)
Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Nagel, R.: One-Parameter Semigroups of Positive Operators. Springer, Heidelberg (1986)
Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306, 837–852 (1988)
Author information
Authors and Affiliations
Corresponding author
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
About this article
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). https://doi.org/10.1007/s11868-020-00349-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11868-020-00349-9
Keywords
- M/G/1 queueing model with second optional service and server breakdowns
- \(C_0\)-semigroup
- Time-dependent solution
- Dispersive operator
- Eigenvalue