Dynamic analysis of the M/G/1 queueing system with multiple phases of operation

Nurehemaiti Yiming; Nurehemaiti Yiming

doi:10.3934/nhm.2024053

Networks and Heterogeneous Media

2024, Volume 19, Issue 3: 1231-1261. doi: 10.3934/nhm.2024053

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Research article

Dynamic analysis of the M/G/1 queueing system with multiple phases of operation

Nurehemaiti Yiming ^{1,2
,
,}

1.
College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China
2.
Xinjiang Key Laboratory of Applied Mathematics, Xinjiang University, Urumqi 830017, China

Received: 06 September 2024 Revised: 21 October 2024 Accepted: 28 October 2024 Published: 31 October 2024

In this paper, we considered the M/G/1 queueing system with multiple phases of operation. First, we have proven the existence and uniqueness of the time-evolving solution for this queueing system. Second, by calculating the spectral distribution of the system operator, we proved that the solution converged at most strongly to its steady-state (static) solution. We also discussed the compactness of the system's corresponding semigroup. Additionally, we investigated the asymptotic behavior of dynamic indicators. Finally, to demonstrate the exponential convergence of the solution, we conducted some numerical analysis.
- queueing system with multiple phases of operation,
- $ C_0- $semigroup,
- operator spectrum,
- asymptotic behavior,
- time-evolving queue length
Citation: Nurehemaiti Yiming. Dynamic analysis of the M/G/1 queueing system with multiple phases of operation[J]. Networks and Heterogeneous Media, 2024, 19(3): 1231-1261. doi: 10.3934/nhm.2024053

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Abstract

In this paper, we considered the M/G/1 queueing system with multiple phases of operation. First, we have proven the existence and uniqueness of the time-evolving solution for this queueing system. Second, by calculating the spectral distribution of the system operator, we proved that the solution converged at most strongly to its steady-state (static) solution. We also discussed the compactness of the system's corresponding semigroup. Additionally, we investigated the asymptotic behavior of dynamic indicators. Finally, to demonstrate the exponential convergence of the solution, we conducted some numerical analysis.

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