Probabilistic Engineering Mechanics, Volume 86, 2026, 104047, ISSN 0266-8920, DOI: https://doi.org/10.1016/j.probengmech.2026.104047.

Probabilistic Reliability Modeling of Delayed Degradation Systems With Mixed Random Shocks and Discontinuous State Jumps

Hao Lyu¹, Jing Pan¹, Chuanhai Chen², Murat Ozku³ and Michael Pecht<sup>4

1</sup>School of Mechanical Engineering and Automation, Northeastern University, Shenyang, China
²Key Laboratory of CNC Equipment Reliability, Ministry of Education, Jilin University, Changchun, China
³Izmir University of Economics, Department of Mathematics, Izmir, Turkiye
⁴Center for Advanced Life Cycle Engineering, University of Maryland, College Park, MD, USA

For more information about this article and related research, please contact Prof. Michael Pecht.

Abstract:

Many engineering components experience a degradation delay period before observable deterioration becomes dominant. In random shock environments, this delay period may be shortened by external shocks, and the subsequent degradation process can involve discontinuous jumps among damage states. To characterize these coupled effects, this paper develops a two-stage probabilistic reliability model for delayed degradation systems subject to mixed random shocks. The first stage describes the degradation delay period, whose termination can be accelerated by shock-induced damage accumulation or run-shock events. The second stage is formulated as a multi-state degradation process in which shocks may induce non-adjacent state jumps. Based on a mixed-shock mechanism and a finite Markov chain embedding framework, the hard-failure process is analytically modeled. Soft failure is described by a path-dependent linear degradation model with state-dependent degradation rates and shock-induced damage increments. The overall reliability is derived using the law of total probability by integrating the transition-time distribution and conditional survival probabilities in the degradation stage. A case study illustrates the framework and shows that ignoring degradation delay or shock-induced state jumps may lead to biased reliability assessment.

This article is available online for free until November 17, 2026.