Abstract and Applied Analysis

On Eigenvalues of the Generator of a C 0 -Semigroup Appearing in Queueing Theory

Geni Gupur

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 27 February 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Export citation


  • 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