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Networks and Heterogeneous Media

2025, Volume 20, Issue 2: 590-624. doi: 10.3934/nhm.2025026
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Asymptotic behavior of the solution of a stochastic clearing queueing system

  • College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China
  • Published: 29 May 2025

  • In this paper, we investigate the asymptotic behavior of the time-dependent solution for the M/G/1 stochastic clearing queueing system operating in a three-phase environment. The mathematical model of this system is characterized by an infinite set of integro-partial differential equations, with boundary conditions that incorporate integral equations. Initially, we employ probability generating functions to demonstrate that 0 is an eigenvalue of the system operator, possessing a geometric multiplicity of one. Subsequently, by invoking Greiner's boundary perturbation method, we establish that all points on the imaginary axis, with the exception of 0, reside within the resolvent set of the system operator. Furthermore, we highlight that 0 also serves as an eigenvalue of the adjoint operator of the system operator, with a geometric multiplicity of unity. As a result, we conclude that the time-dependent solution of the system converges strongly to its steady-state solution.

    Citation: Nurehemaiti Yiming. Asymptotic behavior of the solution of a stochastic clearing queueing system[J]. Networks and Heterogeneous Media, 2025, 20(2): 590-624. doi: 10.3934/nhm.2025026

    Related Papers:

  • In this paper, we investigate the asymptotic behavior of the time-dependent solution for the M/G/1 stochastic clearing queueing system operating in a three-phase environment. The mathematical model of this system is characterized by an infinite set of integro-partial differential equations, with boundary conditions that incorporate integral equations. Initially, we employ probability generating functions to demonstrate that 0 is an eigenvalue of the system operator, possessing a geometric multiplicity of one. Subsequently, by invoking Greiner's boundary perturbation method, we establish that all points on the imaginary axis, with the exception of 0, reside within the resolvent set of the system operator. Furthermore, we highlight that 0 also serves as an eigenvalue of the adjoint operator of the system operator, with a geometric multiplicity of unity. As a result, we conclude that the time-dependent solution of the system converges strongly to its steady-state solution.



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