Point Spectra of the Operator Corresponding to the $M/M/1$ Queueing Model with Working Vacation and Vacation Interruption
Received:October 05, 2017  Revised:November 08, 2018
Key Word: $M/M/1$ queueing model   working vacation and vacation interruption   $C_0$-semigroup   eigenvalue   essential spectral bound  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11801485).
Author NameAffiliation
Ehmet KASIM College of Mathematics and Systems Science, Xinjiang University, Xinjiang 830046, P. R. China 
Geni GUPUR College of Mathematics and Systems Science, Xinjiang University, Xinjiang 830046, P. R. China 
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Abstract:
      In this paper, we consider point spectra of the operator corresponding to the $M/M/1$ queueing model with working vacation and vacation interruption. We prove that the underlying operator has uncountable eigenvalues on the left real line and these results describe the point spectra of the operator. Then, we show that the essential growth bound of the $C_0$-semigroup generated by the operator is 0 and therefore it is not quasi compact, the essential spectral bound of the $C_0$-semigroup is equal to 1. Moreover, our results imply it is impossible that the time-dependent solution of the model exponentially converges to its steady-state solution.
Citation:
DOI:10.3770/j.issn:2095-2651.2019.01.008
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