Analysis of the M/G/1 queueing model with second optional service and server breakdowns

Abstract

We investigate an M/G/1 queue with second optional service and server breakdowns, which is described by infinitely many partial differential equations with boundary conditions and initial conditions. Firstly, by using \(C_0\)-semigroup theory we prove well-posedness of the system and the system has a unique positive time-dependent solution that satisfies the probability condition. Next, by studying spectral properties of the operator corresponding to the model we prove that all points on the imaginary axis except zero belong to the resolvent set of the operator. Lastly, we prove that zero is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model strongly converges to zero.

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Correspondence to Abdukerim Haji.

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This work was supported by the National Natural Science Foundation of China (Nos: 11361057, 11761066).

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Hoshur, A., Haji, A. Analysis of the M/G/1 queueing model with second optional service and server breakdowns. J. Pseudo-Differ. Oper. Appl. 11, 1265–1287 (2020). https://doi.org/10.1007/s11868-020-00349-9

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Keywords

  • M/G/1 queueing model with second optional service and server breakdowns
  • \(C_0\)-semigroup
  • Time-dependent solution
  • Dispersive operator
  • Eigenvalue

Mathematics Subject Classification

  • 47D06
  • 90B25