Evaluating Failure Time Probabilities for Compound Degradation with Linear Path and Mixed Jumps
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Published
Jan 13, 2026
Shihao Cao
Zhihua Wang
Xiangmin Ouyang
Pengjun Zeng
Abstract
Many devices may experience nature degradation and mixed jumps simultaneously whose types can be divided into positive jumps and negative jumps, while these complicated performance rules also bring difficulties in lifetime analysis within the concept of the first hitting time. To address this issue, this paper first proposes a compound degradation process, which is characterized by linear path and mixed jumps. Then, by adopting the idea that transforms the positive jumps into the threshold, an approximate lifetime solution is derived. Given the realistic application of furnace wall, numerical verification shows that the proposed method can maintain consistency with Monte Carlo simulation, while conspicuous errors exist for existing methods, demonstrating that the proposed method can be regarded as theoretical support for the future studies.
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Keywords
Degradation, Mixed jumps, First hitting time, Analytical lifetime distribution
References
Abate, J., & Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. OSRA Journal on Computing. 7 (1) 36-43. doi:10.1287/ijoc.7.1.36.
Cao, S. H., Wang, Z. H., Wu, Q., Ouyang, Xiangmin., Si, X. S., & Liu, C. R. (2025). Failure time analysis for compound degradation procedures involving linear path and negative jumps. Reliability Engineering & System Safety. 253. doi:10.1016/j.ress.2024.110566.
Charri, R., Briat, O., & Vinassa, J. M. (2014). Capacitance recovery analysis and modelling of supercapacitors during cycling ageing tests. Energy Conversion and Management. 82, 37-45. doi:10.1016/j.enconman.2014.02.051.
Che, H. Y., Zeng, S. K., Guo, J. B., & Wang, Y. (2018). Reliability modeling for dependent competing failure processes with mutually dependent degradation process and shock process. Reliability Engineering & System Safety. 180, 168-178. doi:10.1016/j.ress.2018.07.018.
Fan, M. F., Zeng, Z. Z., Zio., E., & Kang, R. (2017). Modeling dependent competing failure processes with degradation-shock dependence. Reliability Engineering & System Safety. 165, 422-430. doi:10.1016/j.ress.2017.05.004.
Kharoufeh, J. P., Finkelstein, D. E., & Mixon, D. G. (2006). Availability of periodically inspected systems with markovian wear and shocks. Journal of Applied Probability. 43 (2), 303-317. doi:10.1239/jap/1152413724.
Kong, X. F., Yang, J., & Li, L. (2021). Remaining useful life prediction for degrading systems with random shocks considering measurement uncertainty. Journal of Manufacturing Systems. 61, 782-798. doi:10.1016/j.jmsy.2021.05.019.
Li, J. X., Wang, Z. H., Zhang, Y. B., Fu, H. M., Liu, C. R., & Krishnaswamy, S. (2017). Degradation data analysis based on a generalized Wiener process subject to measurement error. Mechanical Systems and Signal Processing. 94, 57-72. doi:10.1016/j.ymssp.2017.02.031.
Li, N. P., Gebraeel, N., Lei, Y. G., Bian, L. K., & Si, X. S. (2019). Remaining useful life prediction of machinery under time-varying operating conditions based on a two-factor state-space model. Reliability Engineering & System Safety. 186, 88-100. doi:10.1016/j.ress.2019.02.017.
Peng, H., Feng, Q. M., & Coit, D. W. (2011). Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes. IIE Transactions. 43 (1), 12-22. doi:10.1080/0740817X.2010.491502.
Sun, C., Qu, A., Zhang, J., Shi, Q. Y., & Jia, Z. H. (2023). Remaining useful life prediction for lithium-ion batteries based on improved variational mode decomposition and machine learning algorithm. Energies, 16 (1). doi:10.3390/en16010313.
Tanner, D. M., Walraven, J. A., Helgesen, K., Irwin, L. W., Brown, F., Smith, N. F., & Masters, N. (2000). MEMS reliability in shock environments. 2000 IEEE International Reliability Physics Symposium Proceedings (129-138), April 10-13, San Jose, CA, USA. doi:10.1109/RELPHY.2000.843903.
Wang, H. W., Xiang, J. W., Zhao, X. F., & Li, Y. L. (2019). A kinematic precision reliability evaluation method for rotor-bearing systems considering multi-source wear degradations and random errors. The International Journal of Advanced Manufacturing Technology. 124 (11-12), 4159-4173. doi:10.1007/s00170-022-09383-x.
Wang, W. K., & Chen, X. (2023). Piecewise deterministic Markov process for condition-based imperfect maintenance models. Reliability Engineering & System Safety. 236. doi:10.1016/j.ress.2023.109271.
Wang, Z. Q., Hu, C. H., Si, X. S., & Zio, E. (2018). Remaining useful life prediction of degrading systems subjected to imperfect maintenance: Application to draught fans. Mechanical Systems and Signal Processing. 100, 802-813. doi:10.1016/j.ymssp.2017.08.016.
Wei, G. Z., Zhao, X. J., He, S. G., & He, Z. (2019). Reliability modeling with condition-based maintenance for binary-state deteriorating systems considering zoned shock effects. Computers & Industrial Engineering. 130, 282-297. doi:10.1016/j.cie.2019.02.034.
Xu, Z. C., Xie, N. M., & Li, K. L. (2024). Remaining useful life prediction for lithium-ion batteries with an improved grey particle filter model. Journal of Energy Storage, 78. doi:10.1016/j.est.2023.110081.
Zhang, J. X., Hu, C. H., He, X., Si, X. S., Liu, Y., & Zhou, D. Y. (2017). Lifetime prognostics for deteriorating systems with time-varying random jumps. Reliability Engineering & System Safety. 167, 338-350. doi:10.1016/j.ress.2017.05.047.
Zhang, Z. H, Zhao, C. W, Zhao, Z. P., Wang, F. M., & Zhao, B. (2023). Structural fatigue reliability evaluation based on probability analysis of the number of zero-crossings of stochastic response process. Engineering Failure Analysis. 143. doi:10.1016/j.engfailanal.2022.106923.
Cao, S. H., Wang, Z. H., Wu, Q., Ouyang, Xiangmin., Si, X. S., & Liu, C. R. (2025). Failure time analysis for compound degradation procedures involving linear path and negative jumps. Reliability Engineering & System Safety. 253. doi:10.1016/j.ress.2024.110566.
Charri, R., Briat, O., & Vinassa, J. M. (2014). Capacitance recovery analysis and modelling of supercapacitors during cycling ageing tests. Energy Conversion and Management. 82, 37-45. doi:10.1016/j.enconman.2014.02.051.
Che, H. Y., Zeng, S. K., Guo, J. B., & Wang, Y. (2018). Reliability modeling for dependent competing failure processes with mutually dependent degradation process and shock process. Reliability Engineering & System Safety. 180, 168-178. doi:10.1016/j.ress.2018.07.018.
Fan, M. F., Zeng, Z. Z., Zio., E., & Kang, R. (2017). Modeling dependent competing failure processes with degradation-shock dependence. Reliability Engineering & System Safety. 165, 422-430. doi:10.1016/j.ress.2017.05.004.
Kharoufeh, J. P., Finkelstein, D. E., & Mixon, D. G. (2006). Availability of periodically inspected systems with markovian wear and shocks. Journal of Applied Probability. 43 (2), 303-317. doi:10.1239/jap/1152413724.
Kong, X. F., Yang, J., & Li, L. (2021). Remaining useful life prediction for degrading systems with random shocks considering measurement uncertainty. Journal of Manufacturing Systems. 61, 782-798. doi:10.1016/j.jmsy.2021.05.019.
Li, J. X., Wang, Z. H., Zhang, Y. B., Fu, H. M., Liu, C. R., & Krishnaswamy, S. (2017). Degradation data analysis based on a generalized Wiener process subject to measurement error. Mechanical Systems and Signal Processing. 94, 57-72. doi:10.1016/j.ymssp.2017.02.031.
Li, N. P., Gebraeel, N., Lei, Y. G., Bian, L. K., & Si, X. S. (2019). Remaining useful life prediction of machinery under time-varying operating conditions based on a two-factor state-space model. Reliability Engineering & System Safety. 186, 88-100. doi:10.1016/j.ress.2019.02.017.
Peng, H., Feng, Q. M., & Coit, D. W. (2011). Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes. IIE Transactions. 43 (1), 12-22. doi:10.1080/0740817X.2010.491502.
Sun, C., Qu, A., Zhang, J., Shi, Q. Y., & Jia, Z. H. (2023). Remaining useful life prediction for lithium-ion batteries based on improved variational mode decomposition and machine learning algorithm. Energies, 16 (1). doi:10.3390/en16010313.
Tanner, D. M., Walraven, J. A., Helgesen, K., Irwin, L. W., Brown, F., Smith, N. F., & Masters, N. (2000). MEMS reliability in shock environments. 2000 IEEE International Reliability Physics Symposium Proceedings (129-138), April 10-13, San Jose, CA, USA. doi:10.1109/RELPHY.2000.843903.
Wang, H. W., Xiang, J. W., Zhao, X. F., & Li, Y. L. (2019). A kinematic precision reliability evaluation method for rotor-bearing systems considering multi-source wear degradations and random errors. The International Journal of Advanced Manufacturing Technology. 124 (11-12), 4159-4173. doi:10.1007/s00170-022-09383-x.
Wang, W. K., & Chen, X. (2023). Piecewise deterministic Markov process for condition-based imperfect maintenance models. Reliability Engineering & System Safety. 236. doi:10.1016/j.ress.2023.109271.
Wang, Z. Q., Hu, C. H., Si, X. S., & Zio, E. (2018). Remaining useful life prediction of degrading systems subjected to imperfect maintenance: Application to draught fans. Mechanical Systems and Signal Processing. 100, 802-813. doi:10.1016/j.ymssp.2017.08.016.
Wei, G. Z., Zhao, X. J., He, S. G., & He, Z. (2019). Reliability modeling with condition-based maintenance for binary-state deteriorating systems considering zoned shock effects. Computers & Industrial Engineering. 130, 282-297. doi:10.1016/j.cie.2019.02.034.
Xu, Z. C., Xie, N. M., & Li, K. L. (2024). Remaining useful life prediction for lithium-ion batteries with an improved grey particle filter model. Journal of Energy Storage, 78. doi:10.1016/j.est.2023.110081.
Zhang, J. X., Hu, C. H., He, X., Si, X. S., Liu, Y., & Zhou, D. Y. (2017). Lifetime prognostics for deteriorating systems with time-varying random jumps. Reliability Engineering & System Safety. 167, 338-350. doi:10.1016/j.ress.2017.05.047.
Zhang, Z. H, Zhao, C. W, Zhao, Z. P., Wang, F. M., & Zhao, B. (2023). Structural fatigue reliability evaluation based on probability analysis of the number of zero-crossings of stochastic response process. Engineering Failure Analysis. 143. doi:10.1016/j.engfailanal.2022.106923.
Section
Regular Session Papers
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