On the remaining lifespan of devices

Stochastic Models in Probability and Statistics

دوره 1، شماره 1، مهر 2024، صفحه 105-118   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.22067/smps.2024.45416

نویسنده

Mario Lefebvre^*

Department of Mathematics and Industrial Engineering, Polytechnique Montreal, Montreal, Canada

چکیده

Let X(t) denote the remaining lifespan of a device. Two-dimensional degenerate diffusion processes (X(t),Y(t)), where Y(t) is a variable that influences the remaining lifespan, are proposed to model the evolution of X(t) over time. These processes are defined in such a way that X(t) will be strictly decreasing as time increases: dX(t) = rho[X(t),Y(t)] dt, where rho is a strictly negative function and {Y(t), t > 0} is a diffusion process. Next, optimal control problems for such diffusion processes, in which the final time is the random time when the device is considered to be worn out, are considered. This type of problems is known as homing problems. The dynamic programming equation satisfied by the value function is derived and particular problems are solved explicitly for diffusion processes {Y(t), t > 0} that are important for applications. To do so, we must solve non-linear partial differential equations, subject to the appropriate boundary conditions.

کلیدواژه‌ها

‎Brownian motion؛ ‎ Degradation؛ ‎ Diffusion processes‎؛ ‎First-passage time؛ ‎ ‎Optimal control

مراجع

1.  Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions with Formulas,
   Graphs, and Mathematical Tables. Dover, New York.
2. Ghamlouch, H., Grall, A. and Fouladirad, M. (2015). On the use of jump-diffusion process for maintenance decision-making: A first step. In 2015 Annual Reliability and Maintainability Symposium (RAMS), 1–6.
3. Kannan, D. (1979). An Introduction to Stochastic Processes. North Holland, New York.
4. Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
5. Lefebvre, M. and Gaspo, J. (1996a). Optimal control of wear processes. IEEE Transactions on Automatic Control, 41, 112–115.
6. Lefebvre, M. and Gaspo, J. (1996b). Controlled wear processes: a risk-sensitive formulation. Engineering Optimization, 26, 187–194.
7.  Lefebvre, M. (2000). Risk-averse control of a three-dimensional wear process. International Journal of Control, 73, 1307–1311.
8. Lefebvre, M. (2007). Applied Stochastic Processes. Springer, New York.
9. Lefebvre, M. (2010). Mean fist-passage time to zero for wear processes. Stochastic Models, 26, 46–53.
10. Makasu, C. (2022). Homing problems with control in the diffusion coefficent. IEEE Transactions on Automatic Control, 67, 3770–3772.
11. Nicolai, R. P. and Dekker, R. (2007). A comparison of two non-stationary degradation processes. In: Aven, T. and Vinnem, J. E. (eds.) Risk, Reliability and Societal Safety: Proceedings ESREL 2007, 25-27 June, Stavanger, Taylor & Francis, London, 1109–1114.
12. Øksendal, B. (2003). Stochastic Diffrential Equations: An Introduction with Applications. 6th Ed., Springer-Verlag, Berlin.
13. Rishel, R. (1991). Controlled wear processes: modeling optimal control. IEEE Transactions on Automatic Control, 36, 1100–1102. Shahraki, A. F., Yadav, O. P. and Liao, H. (2017). A review on degradation modelling and its engineering applications. International Journal of Performability Engineering, 13, 299–314.
14. Whittle, P. (1982). Optimization Over Time I. Wiley, Chichester.
15. Whittle, P, (1990) Risk-Sensitive Optimal Control. Wiley, Chichester.
16. Whitmore, G. A. and Schenkelberg, F, (1997). Modelling accelerated degradation data using Wiener diffusion with a time scale transformation. Lifetime Data Analysis, 3, 27–45.
17. Ye, Z.-S., Wang, Y., Tsui, K.-L. and Pecht, M. (2013). Degradation data analysis using Wiener processes with measurement errors. IEEE Transactions on Reliability, 62, 772–780.
18. Zhai, Q., Chen, P., Hong, L. and Shen, L. (2018) A random-effects Wiener degradation model based on accelerated failure time. Reliability Engineering & System Safety, 180, 94–103.
19. Zhang, Z., Hu, C., Si, X., Zhang, J. and Zheng, J. (2017) Stochastic degradation process modeling and remaining useful life estimation with flexible random-effects. Journal of the Franklin Institute, 354, 2477–2499.
20. Zhou, S., Tang, Y. and Xu, A. (2021). A generalized Wiener process with dependent degradation rate and volatility and time-varying mean-to-variance ratio. Reliability Engineering & System Safety, 216, 107895.