Dynamic analysis of Gnedenko system with multiple working vacations

Ehmet Kasim; Rena Eskar; Ehmet Kasim; Rena Eskar

doi:10.3934/nhm.2026084

Networks and Heterogeneous Media

2026, Volume 21, Issue 6: 2246-2285. doi: 10.3934/nhm.2026084

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Research article

Dynamic analysis of Gnedenko system with multiple working vacations

Ehmet Kasim ^,,
Rena Eskar

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

Received: 14 July 2026 Revised: 22 September 2026 Accepted: 19 October 2026 Published: 30 October 2026

We present a dynamic analysis of the Gnedenko repairable system with multiple working vacations, which is comprised of $ N $ operating units and one warm standby unit. The system dynamics are governed by partial differential equations with integral boundary conditions. By applying the $ C_0- $semigroup theory, we establish the well-posedness of the system and the strong convergence of the time-dependent solution to its steady-state solution. Numerical examples illustrate the effect of key parameters on time-dependnet reliability indices, providing practical insights into the system performance.
- Gnedenko system,
- multiple working vacations,
- $ C_0- $semigroup,
- dynamic solution,
- asymptotic behavior
Citation: Ehmet Kasim, Rena Eskar. Dynamic analysis of Gnedenko system with multiple working vacations[J]. Networks and Heterogeneous Media, 2026, 21(6): 2246-2285. doi: 10.3934/nhm.2026084

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Abstract

We present a dynamic analysis of the Gnedenko repairable system with multiple working vacations, which is comprised of $ N $ operating units and one warm standby unit. The system dynamics are governed by partial differential equations with integral boundary conditions. By applying the $ C_0- $semigroup theory, we establish the well-posedness of the system and the strong convergence of the time-dependent solution to its steady-state solution. Numerical examples illustrate the effect of key parameters on time-dependnet reliability indices, providing practical insights into the system performance.

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