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On eigenvalues of the generator of a \(C_0\)-semigroup appearing in queueing theory. (English) Zbl 1473.47015

Summary: 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.

MSC:

47D06 One-parameter semigroups and linear evolution equations
90B22 Queues and service in operations research
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[1] Cox, D. R., The analysis of non-Markovian stochastic process by the inclusion of supplementary variables, Proceedings of Cambridge Philosophical Society, 55, 433-441, (1955) · Zbl 0067.10902
[2] Gupur, G.; Li, X. Z.; Zhu, G. T., Functional Analysis Method in Queueing Theory, (2001), Hertfordshire, UK: Research Information Limited, Hertfordshire, UK · Zbl 0971.60097
[3] Gupur, G., Advances in queueing models’ research, Acta Analysis Functionalis Applicata, 13, 3, 225-245, (2011) · Zbl 1249.90001
[4] Gupur, G., Functional Analysis Methods for Reliability Models, (2011), Basel, Switzerland: Springer, Basel, Switzerland · Zbl 1223.90004
[5] Radl, A., Semigroups applied to transport and queueing processes [Ph.D. thesis], (2006), Tübingen, Germany: Eberhard Karls Universität Tübingen, Tübingen, Germany
[6] Zhang, L.; Gupur, G., Another eigenvalue of the \(M / M / 1\) operator, Acta Analysis Functionalis Applicata, 10, 1, 81-91, (2008) · Zbl 1174.90414
[7] Kasim, E.; Gupur, G., Other eigenvalues of the M/M/1 operator, Acta Analysis Functionalis Applicata, 13, 45-53, (2011) · Zbl 1249.35235
[8] Engel, K. J.; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, (2000), New York, NY, USA: Springer, New York, NY, USA · Zbl 0952.47036
[9] Song, J.; Yu, J. Y., Population System Control, (1988), Berlin, Germany: Springer, Berlin, Germany
[10] Webb, G. F., Theory of Nonlinear Age-Dependent Population Dynamics, (1985), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0555.92014
[11] Xu, H. B.; Hu, W. W., Modelling and analysis of repairable systems with preventive maintenance, Applied Mathematics and Computation, 224, 46-53, (2013) · Zbl 1334.90046
[12] Zhao, Z. X.; Shao, C.; Xu, G. Q., Spectral analysis of an operator in the \(M / M / 1\) queueing model described by ordinary differential equations, Acta Analysis Functionalis Applicata, 12, 2, 186-192, (2010) · Zbl 1140.47333
[13] Nagel, R., One-Parameter Semigroups of Positive Operators, (1986), Berlin, Germany: Springer, Berlin, Germany · Zbl 0585.47030
[14] Browder, F. E., On the spectral theory of elliptic differential operators I, Mathematische Annalen, 142, 22-130, (1960-1961) · Zbl 0104.07502
[15] Kato, T., Perturbation Theory for Linear Operators, (1976), New York, NY, USA: Springer, New York, NY, USA
[16] Schechter, M., Essential spectra of elliptic partial differential equations, Bulletin of the American Mathematical Society, 73, 567-572, (1967) · Zbl 0153.45501
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