Well-Posedness and Asymptotic Behavior of the Standby Repairable System with Partial Failure and Priority
- 1College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China
Abstract
This paper investigates the well-posedness (WP) and asymptotic stability of a standby repairable system with partial failure and priority, wherein components in a partial failure state remain unrepaired during system operation. The governing integro-differential equations are reformulated as an abstract Cauchy problem (ACP) on an appropriately chosen Banach space via the introduction of a linear operator. Using C0−semigroup theory together with spectral analysis, we prove that the system operator generates a positive contraction C0−semigroup, thereby guaranteeing the existence of unique time-dependent solution (TDS). Moreover, through a detailed examination of the spectral distribution of the generator on the imaginary axis, we establish the strong convergence of the TDS to its steady-state solution (SSS) as time tends to infinity. Complementing these analytical results, a Monte Carlo simulation (MCS) is performed to track the transient evolution of reliability indices, providing numerical validation of the theoretical predictions. These results offer a theoretical basis for evaluating system reliability and inform maintenance strategies in continuous-production environments.
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BibTeX
@article{kasim2026wellposedness,
title = {{Well-Posedness and Asymptotic Behavior of the Standby Repairable System with Partial Failure and Priority}},
author = {Ehmet Kasim and Rena Eskar and Taiyong Wang},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203780},
url = {https://www.mdpi.com/2227-7390/14/20/3780},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Well-Posedness and Asymptotic Behavior of the Standby Repairable System with Partial Failure and Priority
AU - Ehmet Kasim
AU - Rena Eskar
AU - Taiyong Wang
PY - 2026
DA - 2026/10/18
JO - Mathematics
DO - 10.3390/math14203780
UR - https://www.mdpi.com/2227-7390/14/20/3780
AB - This paper investigates the well-posedness (WP) and asymptotic stability of a standby repairable system with partial failure and priority, wherein components in a partial failure state remain unrepaired during system operation. The governing integro-differential equations are reformulated as an abstract Cauchy problem (ACP) on an appropriately chosen Banach space via the introduction of a linear operator. Using C0−semigroup theory together with spectral analysis, we prove that the system operator generates a positive contraction C0−semigroup, thereby guaranteeing the existence of unique time-dependent solution (TDS). Moreover, through a detailed examination of the spectral distribution of the generator on the imaginary axis, we establish the strong convergence of the TDS to its steady-state solution (SSS) as time tends to infinity. Complementing these analytical results, a Monte Carlo simulation (MCS) is performed to track the transient evolution of reliability indices, providing numerical validation of the theoretical predictions. These results offer a theoretical basis for evaluating system reliability and inform maintenance strategies in continuous-production environments.
SN - 2227-7390
ER -