Parallel–series systems are fundamental in many industrial and engineering applications, yet their reliability assessment and improvement remain challenging, particularly when components exhibit non-constant failure rates. This study addresses this challenge by modeling a hybrid parallel–series system whose components follow the Akshaya lifetime distribution, a flexible model that can capture various hazard-rate shapes. For this system, we derive closed-form analytical expressions for key reliability indices, including the system reliability function, mean time to failure (MTTF), reliability equivalence factors (REFs), and δ-fractiles. To enhance system performance, four improvement strategies are formulated and analytically compared: failure-rate reduction, hot duplication, cold duplication with a perfect switch, and cold duplication with an imperfect switch. A comprehensive numerical case study validates the theoretical derivations and demonstrates the effectiveness of each strategy. The results show that cold duplication with a perfect switch yields the highest reliability gain, and the computed REFs provide a quantitative tool for balancing redundancy against component-level improvements. This work provides reliability engineers with a comprehensive analytical framework for the design and enhancement of complex parallel-series systems.
Ramadan et al. (Wed,) studied this question.
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