Well-posedness and asymptotic behavior of the time-dependent solution of an M/G/1 queueing model

  • Nurehemaiti Yiming
  • Geni GupurEmail author


By using the Hille–Yosida theorem, Phillips theorem and Fattorini theorem we prove that the M/G/1 queueing model with vacations and multiple phases of operation, which is described by infinitely many partial differential equations with integral boundary conditions, has a unique positive time-dependent solution that satisfies the probability condition. Next, by studying the spectrum of the operator, which corresponds to the model, on the imaginary axis we prove that the time-dependent solution of the model strongly converges to its steady-state solution.


M/G/1 queueing model with vacations and multiple phases of operation Time-dependent solution \(C_0\)-semigroup Resolvent set Eigenvalue 

Mathematics Subject Classification

Primary 47D03 47A10 Secondary 60K25 


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceXinjiang UniversityÜrümqiPeople’s Republic of China

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