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<p>This paper develops an analytic solution framework for renewal equations arising from lifetime distributions with polynomial cumulative hazard functions. The approach builds on a recently developed inference framework for such models, enabling a consistent transition from parameter estimation to dynamic reliability analysis. Recasting the renewal equation using a zero-mass kernel yields a constructive representation of the renewal density. We show that the solution admits a decomposition into an explicit baseline term and a finite polynomial correction multiplied by the survival function. The correction coefficients are obtained from a lower-triangular linear system, providing a computable alternative to classical series expansions. A uniform error bound with stretched-exponential decay is established, together with a principled criterion for selecting the polynomial order. A case study involving circuit breaker reliability, based on Weibull parameters reported in the literature, demonstrates that the analytic approximation <b>closely matches numerical</b> convolution-based solutions while enabling efficient evaluation in maintenance cost calculations. The proposed framework provides a tractable analytic alternative to numerical methods and supports repeated evaluation scenarios arising in reliability analysis and maintenance planning.</p>

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Analytic renewal equation solutions for polynomial-hazard models: explicit construction and error bounds

  • 期刊论文

  • 发表日期：

    2026年5月27日

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  • Serguei Maximov,
  • Jose G. Tirado-Serrato,
  • Margarita J. Toledo-Vazquez,
  • Alfredo Sanchez

摘要

This paper develops an analytic solution framework for renewal equations arising from lifetime distributions with polynomial cumulative hazard functions. The approach builds on a recently developed inference framework for such models, enabling a consistent transition from parameter estimation to dynamic reliability analysis. Recasting the renewal equation using a zero-mass kernel yields a constructive representation of the renewal density. We show that the solution admits a decomposition into an explicit baseline term and a finite polynomial correction multiplied by the survival function. The correction coefficients are obtained from a lower-triangular linear system, providing a computable alternative to classical series expansions. A uniform error bound with stretched-exponential decay is established, together with a principled criterion for selecting the polynomial order. A case study involving circuit breaker reliability, based on Weibull parameters reported in the literature, demonstrates that the analytic approximation closely matches numerical convolution-based solutions while enabling efficient evaluation in maintenance cost calculations. The proposed framework provides a tractable analytic alternative to numerical methods and supports repeated evaluation scenarios arising in reliability analysis and maintenance planning.