## Abstract and Applied Analysis

### On Eigenvalues of the Generator of a ${C}_{0}$-Semigroup Appearing in Queueing Theory

Geni Gupur

#### Abstract

We describe the point spectrum of the generator of a ${C}_{0}$-semigroup associated with the M/M/1 queueing model that is governed by an infinite system of partial differential equations with integral boundary conditions. Our results imply that the essential growth bound of the ${C}_{0}$-semigroup is 0 and, therefore, that the semigroup is not quasi-compact. Moreover, our result also shows that it is impossible that the time-dependent solution of the M/M/1 queueing model exponentially converges to its steady-state solution.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 896342, 9 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425047823

Digital Object Identifier
doi:10.1155/2014/896342

Mathematical Reviews number (MathSciNet)
MR3272224

Zentralblatt MATH identifier
07023264

#### Citation

Gupur, Geni. On Eigenvalues of the Generator of a ${C}_{0}$ -Semigroup Appearing in Queueing Theory. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 896342, 9 pages. doi:10.1155/2014/896342. https://projecteuclid.org/euclid.aaa/1425047823

#### References

• D. R. Cox, “The analysis of non-Markovian stochastic process by the inclusion of supplementary variables,” Proceedings of Cambridge Philosophical Society, vol. 55, pp. 433–441, 1955.
• G. Gupur, X. Z. Li, and G. T. Zhu, Functional Analysis Method in Queueing Theory, Research Information Limited, Hertfordshire, UK, 2001.
• G. Gupur, “Advances in queueing models' research,” Acta Ana-lysis Functionalis Applicata, vol. 13, no. 3, pp. 225–245, 2011.
• G. Gupur, Functional Analysis Methods for Reliability Models, Springer, Basel, Switzerland, 2011.
• A. Radl, Semigroups applied to transport and queueing processes [Ph.D. thesis], Eberhard Karls Universität Tübingen, Tübingen, Germany, 2006.
• L. Zhang and G. Gupur, “Another eigenvalue of the $M/M/1$ operator,” Acta Analysis Functionalis Applicata, vol. 10, no. 1, pp. 81–91, 2008.
• E. Kasim and G. Gupur, “Other eigenvalues of the M/M/1 opera-tor,” Acta Analysis Functionalis Applicata, vol. 13, pp. 45–53, 2011.
• K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, NY, USA, 2000.
• J. Song and J. Y. Yu, Population System Control, Springer, Berlin, Germany, 1988.
• G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, NY, USA, 1985.
• H. B. Xu and W. W. Hu, “Modelling and analysis of repairable systems with preventive maintenance,” Applied Mathematics and Computation, vol. 224, pp. 46–53, 2013.
• Z. X. Zhao, C. Shao, and G. Q. Xu, “Spectral analysis of an operator in the $M/M/1$ queueing model described by ordinary differential equations,” Acta Analysis Functionalis Applicata, vol. 12, no. 2, pp. 186–192, 2010.
• R. Nagel, One-Parameter Semigroups of Positive Operators, Springer, Berlin, Germany, 1986.
• F. E. Browder, “On the spectral theory of elliptic differential ope-rators I,” Mathematische Annalen, vol. 142, pp. 22–130, 1960-1961.
• T. Kato, Perturbation Theory for Linear Operators, Springer, New York, NY, USA, 2nd edition, 1976.
• M. Schechter, “Essential spectra of elliptic partial differential equations,” Bulletin of the American Mathematical Society, vol. 73, pp. 567–572, 1967. \endinput