Hostname: page-component-669899f699-tpknm Total loading time: 0 Render date: 2025-04-25T12:41:14.570Z Has data issue: false hasContentIssue false
  • English
  • Français

Reliability and optimal age replacement policy of a system subject to shocks following a Markovian arrival process

Part of: Operations research and management science Special processes Stochastic processes

Published online by Cambridge University Press:  14 February 2025

Dheeraj Goyal Open the ORCID record for Dheeraj Goyal [Opens in a new window] ,
Min Xie  and
Min Gong
Dheeraj Goyal*
Affiliation:
Centre for Intelligent Multidimensional Data Analysis Limited and Indian Institute of Technology Kanpur
Min Xie*
Affiliation:
Centre for Intelligent Multidimensional Data Analysis Limited and City University of Hong Kong
Min Gong*
Affiliation:
Jinan University
*
*Postal address: Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Uttar Pradesh-208016, India. Email: [email protected]
**Postal address: Centre for Intelligent Multidimensional Data Analysis Limited, Hong Kong Science Park, Hong Kong.
***Postal address: College of Information Science and Technology, Jinan University, Guangzhou, China
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper defines and studies a broad class of shock models by assuming that a Markovian arrival process models the arrival pattern of shocks. Under the defined class, we show that the system’s lifetime follows the well-known phase-type distribution. Further, we examine the age replacement policy for systems with a continuous phase-type distribution, identifying sufficient conditions for determining the optimal replacement time. Since phase-type distributions are dense in the class of lifetime distributions, our findings for the age replacement policy are widely applicable. We include numerical examples and graphical illustrations to support our results.

Keywords

Reliabilitymean residual lifetimeage replacementMarkovian arrival processshock models

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Secondary: 60G55: Point processes 60K10: Applications (reliability, demand theory, etc.)
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 18
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Asmussen, S. (2000). Matrix-analytic models and their analysis. Scand. J. Statist. 27, 193–226.CrossRefGoogle Scholar
Bladt, M. and Nielsen, B. (2017). Matrix-Exponential Distributions in Applied Probability. Springer, New York.CrossRefGoogle Scholar
Bozbulut, A. R. and Eryilmaz, S. (2020). Generalized extreme shock models and their applications. Commun. Statist. Sim. Comput. 49, 110–120.CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2016). New shock models based on the generalized Pólya process. Europ. J. Operat. Res. 251, 135–141.CrossRefGoogle Scholar
Chakravarthy, S. R. (2012). Maintenance of a deteriorating single server system with Markovian arrivals and random shocks. Europ. J. Operat. Res. 222, 508–522.CrossRefGoogle Scholar
Dizbin, N. M. and Tan, B. (2020). Optimal control of production-inventory systems with correlated demand inter-arrival and processing times. Int. J. Prod. Econom. 228, 107692.CrossRefGoogle Scholar
Eryilmaz, S. (2017). Computing optimal replacement time and mean residual life in reliability shock models. Comput. Ind. Eng. 103, 40–45.CrossRefGoogle Scholar
Eryilmaz, S. (2021). Revisiting discrete time age replacement policy for phase-type lifetime distributions. Europ. J. Operat. Res. 295, 699–704.CrossRefGoogle Scholar
Eryilmaz, S. and Kan, C. (2019). Reliability and optimal replacement policy for an extreme shock model with a change point. Reliability Eng. System Safety, 190, 106513.CrossRefGoogle Scholar
Eryilmaz, S. and Tekin, M. (2019). Reliability evaluation of a system under a mixed shock model. J. Comput. Appl. Math. 352, 255–261.CrossRefGoogle Scholar
Gong, M., Xie, M. and Yang, Y. (2018). Reliability assessment of system under a generalized run shock model. J. Appl. Prob. 55, 1249–1260.CrossRefGoogle Scholar
Goyal, D., Ali, R. and Hazra, N. K. (2024). Reliability analysis of $\delta$ -shock models based on the Markovian arrival process. Appl. Stochast. Models Business Industry, 40, 1291–1312. DOI: 10.1002/asmb.2858.CrossRefGoogle Scholar
Goyal, D., Hazra, N. K. and Finkelstein, M. (2022). On the general $\delta$ -shock model. TEST 31, 994–1029.CrossRefGoogle Scholar
Goyal, D., Hazra, N. K. and Finkelstein, M. (2023). A general class of shock models with dependent inter-arrival times. TEST 32, 1079–1105.CrossRefGoogle Scholar
Goyal, D., Hazra, N. K. and Finkelstein, M. (2024). Shock models based on renewal processes with matrix Mittag–Leffler distributed inter-arrival times. J. Comput. Appl. Math. 435, 115090.CrossRefGoogle Scholar
Gut, A. and Hüsler, J. (1999). Extreme shock models. Extremes 2, 295–307.CrossRefGoogle Scholar
He, Q. M. (2014). Fundamentals of Matrix-Analytic Methods. Springer, London.CrossRefGoogle Scholar
Johnson, C. R., Smith, R. L. and Tsatsomeros, M. J. (2020). Matrix Positivity. Cambridge University Press.CrossRefGoogle Scholar
Landriault, D. and Shi, T. (2015). Occupation times in the MAP risk model. Insurance Math. Econom. 60, 75–82.CrossRefGoogle Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Latouche, G., Remiche, M. A. and Taylor, P. (2003). Transient Markov arrival processes. Ann. Appl. Prob. 13, 6280–640.CrossRefGoogle Scholar
Levitin, G., Finkelstein, M. and Dai, Y. (2018). Heterogeneous standby systems with shocks-driven preventive replacements. Europ. J. Operat. Res. 266, 1189–1197.CrossRefGoogle Scholar
Lyu, H., Qu, H., Ma, L., Wang, S. and Yang, Z. (2022). Reliability assessment of a system with multi-shock sources subject to dependent competing failure processes under phase-type distribution. Quality Reliability Eng. Int. 38, 2820–2844.CrossRefGoogle Scholar
Mallor, F. and Omey, E. (2001). Shocks, runs and random sums. J. Appl. Prob. 38, 438–448.CrossRefGoogle Scholar
Montoro-Cazorla, D. and Pérez-Ocón, R. (2011). Two shock and wear systems under repair standing a finite number of shocks. Europ. J. Operat. Res. 214, 298–307.CrossRefGoogle Scholar
Montoro-Cazorla, D. and Pérez-Ocón, R. (2014). A reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K . Europ. J. Operat. Res. 235, 636–642.CrossRefGoogle Scholar
Montoro-Cazorla, D. and Pérez-Ocón, R. (2015). A reliability system under cumulative shocks governed by a BMAP. Appl. Math. Modelling 39, 7620–7629.CrossRefGoogle Scholar
Nakagawa, T. (2005). Maintenance Theory of Reliability. Springer, London.Google Scholar
Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16, 764–779.CrossRefGoogle Scholar
Ozkut, M. (2023). Reliability and optimal replacement policy for a generalized mixed shock model. TEST 32, 1038–1054. DOI: 10.1007/s11749-023-00864-z.CrossRefGoogle Scholar
Pérez-Ocón, R. and del Carmen Segovia, M. (2009). Shock models under a Markovian arrival process. Math. Comput. Modelling 50, 879–884.CrossRefGoogle Scholar
Ramírez-Cobo, P., Marzo, X., Olivares-Nadal, A., Alvarez Francoso, J. and Carrizosa, E. (2014). The Markovian arrival process: A statistical model for daily precipitation amounts. J. Hydrology 510, 459–471.CrossRefGoogle Scholar
Safaei, F., Châtelet, E. and Ahmadi, J. (2020). Optimal age replacement policy for parallel and series systems with dependent components. Reliability Eng. System Safety 197, 106798.CrossRefGoogle Scholar
Segovia, M. C. and Labeau, P. E. (2013). Reliability of a multi-state system subject to shocks using phase-type distributions. Appl. Math. Modelling 37, 4883–4904.CrossRefGoogle Scholar
Tank, F. and Eryilmaz, S. (2015). The distributions of sum, minima and maxima of generalized geometric random variables. Statist. Papers 56, 1191–1203.CrossRefGoogle Scholar
Wang, X., Zhao, X., Wu, C. and Wang, S. (2022). Mixed shock model for multi-state weighted k-out-of-n: F systems with degraded resistance against shocks. Reliability Eng. System Safety 217, 108098.CrossRefGoogle Scholar
Yalcin, F., Eryilmaz, S. and Bozbulut, A. R. (2018). A generalized class of correlated run shock models. Dependence Modeling 6, 131–138.CrossRefGoogle Scholar
Zhao, X., Wang, S., Wang, X. and Fan, Y. (2020). Multi-state balanced systems in a shock environment. Reliability Eng. System Safety 193, 106592.CrossRefGoogle Scholar