Spectral distribution and semigroup properties of a queueing model with exceptional service time

Nurehemaiti Yiming; Nurehemaiti Yiming

doi:10.3934/nhm.2024036

Networks and Heterogeneous Media

2024, Volume 19, Issue 2: 800-821. doi: 10.3934/nhm.2024036

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Research article

Spectral distribution and semigroup properties of a queueing model with exceptional service time

Nurehemaiti Yiming ^{1,2
,
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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: 20 June 2024 Revised: 08 August 2024 Accepted: 13 August 2024 Published: 16 August 2024

In this paper, we studied the spectrum and semigroup properties of the M/G/1 queueing model with exceptional service time for the first customer in each busy period. First, we described the point spectrum of the system operator that corresponds to the model, and we prove that the system operator has an uncountable infinite number of eigenvalues on the left-half complex plane. Second, by using the spectrum analysis and semigroup theory, we obtained that the spectrum-determined growth (SDG) condition holds and the semigroup is not asymptotically stable, compact, eventually compact, or even quasi-compact. Finally, in order to clarify the results of spectral distribution, some numerical analysis were conducted.
- queueing model with exceptional service time,
- operator spectrum,
- C₀-semigroup,
- stability,
- compactnecss
Citation: Nurehemaiti Yiming. Spectral distribution and semigroup properties of a queueing model with exceptional service time[J]. Networks and Heterogeneous Media, 2024, 19(2): 800-821. doi: 10.3934/nhm.2024036

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Abstract

In this paper, we studied the spectrum and semigroup properties of the M/G/1 queueing model with exceptional service time for the first customer in each busy period. First, we described the point spectrum of the system operator that corresponds to the model, and we prove that the system operator has an uncountable infinite number of eigenvalues on the left-half complex plane. Second, by using the spectrum analysis and semigroup theory, we obtained that the spectrum-determined growth (SDG) condition holds and the semigroup is not asymptotically stable, compact, eventually compact, or even quasi-compact. Finally, in order to clarify the results of spectral distribution, some numerical analysis were conducted.

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