<![CDATA[Purpose: This study develops a summability-aware workflow that links reliability modelling, Laplace-domain analysis, numerical inversion and constrained optimization for time-dependent reliability and availability.Research Methodology: The framework covers survival and hazard functions, series and parallel structures, a two-state repair model, transform-domain representations, numerical inversion checks and a reliability-cost optimization formulation. Cesàro averaging is treated as a sequence-smoothing and convergence diagnostic rather than a substitute for a valid inverse transform. A worked example uses stated hypothetical rates.Results: For two components with failure rates of 0.0005 and 0.0010 per hour, reliability at 400 hours was 0.5488 in series and 0.9402 in parallel. Fixed-Talbot and de Hoog inversion of the availability transform converged to round-off error, while Gaver-Stehfest converged more slowly, and the lowest-cost design meeting a 0.90 reliability threshold was not the most reliable one.Conclusions: Summability can help interpret oscillatory numerical sequences, but reliable decisions require sound assumptions, independent checks and validation before optimization.Limitations: All rates and design options are hypothetical, and the models assume independence and constant rates.Contributions: The study offers a reproducible verification protocol and a model-to-decision workflow for student research and early-stage industrial reliability studies.]]>
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