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Spectral properties of the \(M^{[X]}/G/1\) operator and its application. (English) Zbl 1217.47142

Summary: By studying spectral properties of the \(M^{[X]}/G/1\) operator which corresponds to the \(M^{[X]}/G/1\) retrial queueing model with server breakdowns and constant rate of repeated attempts, we obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of both the operator and its adjoint operator with geometric multiplicity one. Therefore, by combining these results with our previous result [Theorem 14 of Geni Gupur, Xue-zhi Li and Guangtian Zhu, “Functional Analysis Method in Queueing Theory” (Hemel Hempstead, Hertfordshire: Research Information Ltd.) (2001; per bibl.)], we deduce that the time-dependent solution of the \(M^{[X]}/G/1\) retrial queueing model with server breakdowns and constant rate of repeated attempts strongly converges to its steady-state solution.

MSC:

47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
47D03 Groups and semigroups of linear operators
47A10 Spectrum, resolvent
60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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References:

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