According to the representation of the resolvent set
we obtain that the all points on the imaginary axis except zero belong to the resolvent set of the operator and 0 is an eigenvalue of its adjoint operator with algebraic multiplicity one
then we obtain that the time-dependent solution of this model converges strongly to its steady state solution.
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references
Gautam Choudhury.Some aspects of an M/G/1 queueing system with optional second service[J].Top,2003,11:141-150.
Alim Miji,Geni Gupur.Asymptotic Behavior of the Time-Dependent Solution of the M/M/1 Queueing Model with OptionalSecond Service[J].International Journal of Pure and Applied Mathematics,2011,69:289-328.
Geni Gupur,Li Xue-Zhi,Zhu Guang-tian.Functional Analysis Method in Queueing Theory[M].Hertfordshire:ResearchInformation Ltd,2001.
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