The reliability of a two identical components cold standby system with replaceable repair equipment and multiple failure modes

In engineering systems such as aerospace, nuclear power, and flight control, redundancy design is commonly adopted to ensure system reliability. Cold standby is an important redundancy design, which describes a scenario where standby components remain completely inactive with no performance degradation or failure risk until activated. As technological systems grow increasingly complex, they face diverse failure modes caused by hardware defects, software errors, environmental factors, or human intervention. Consequently, system failures may manifest in a wide variety of forms. In fact, reliability models with multiple failure modes represent a natural extension of traditional binary-state reliability models, providing a more precise and refined characterization of system behavior. However, existing research predominantly assumes systems have only two (e.g., open-circuit/short-circuit) or three (e.g., hardware/software/human-induced) failure modes, with limited studies addressing reliability models incorporating more diverse failure mode classifications. Furthermore, in reliability system studies, when an operational component fails, repair technicians typically utilize specific tools (referred to as repair equipment) to restore the faulty component. During the repair process, the repair equipment itself may fail due to aging, wear and tear, or environmental factors. In such cases, the failed repair equipment must be replaced before the repair of the original faulty component can resume. This scenario represents a more general repairable system framework, and many classical reliability models can be viewed as special cases. Wang et al. [1] investigated a warm standby repairable system consisting of two dissimilar units and a repairman, where unit 1 has priority. Under the assumption that the repair time of failed units follows a general distribution, they employed Markov renewal process theory, the Laplace transforms, and the Cramer’s rule to analyze three key reliability measures, i.e., the mean time to the first failure, the system availability, and the expected number of failures within (0, t]. Furthermore, they examined the effect of system parameters on these steady-state reliability measures. Jiang and Tang [2, 3] introduced the strategies of “replaceable repair equipment” and “delayed repair” respectively into a two identical units parallel repairable system with two types of failure modes, where the repair time of failed units follows a general distribution. By employing the Markov renewal process theory and the Laplace transforms, they analyzed a series of steady-state reliability measures. Wei and Tang [4] investigated a cold standby repairable system with two dissimilar units with replaceable repair equipment and delayed repair. Utilizing Markov renewal process theory, they analyzed several reliability metrics including the steady-state system availability, the mean time to the first failure, the failure frequency, and the probability of waiting for repair. Additionally, numerical examples were provided to illustrate the corrections of theoretical results. Li et al. [5] investigated transient reliability measures for a cold standby system with two dissimilar units featuring priority, imperfect repair, and general repair time distribution. Their analysis focused on reliability function, and distribution function of the number of cycles within (0, t]. Wu et al. [6] examined a k-out-of-n(G) system with replaceable repair equipment, where the repair time of the failed units and the replacement time of the repair equipment obey a general distribution, and the repair equipment has c possible failure states. Using Markov renewal process theory, the authors derived recursive expressions for the steady state system availability, the steady state failure frequency, the mean time to the first failure, the probability of repair equipment being in failed state, as well as its failure frequency. Based on these studies, this paper investigates a cold standby repairable system consisting of two identical units with replaceable repair equipment and the operating component has c failure modes. By employing the Markov renewal process theory and the Laplace-Stieltjes transform, we derive the expressions of system reliability along with some special cases. Additionally, numerical examples are provided