IDEAS home Printed from https://ideas.repec.org/a/eee/reensy/v201y2020ics0951832020304919.html
   My bibliography  Save this article

Geometric-Like Processes: An Overview and Some Reliability Applications

Author

Listed:
  • Arnold, Richard
  • Chukova, Stefanka
  • Hayakawa, Yu
  • Marshall, Sarah

Abstract

The geometric process (GP) can be used to model the occurrence of events, which have an underlying monotonic trend. This type of trend can be observed in many practical problems in reliability, in particular in the recurrent failures of repairable systems. There are also common areas of everyday life, such as epidemiology, business, and health, where such trends in inter-event times can be observed. In order to provide greater flexibility in modelling phenomena and situations involving monotonic trends, a variety of extensions of the geometric process have been proposed. This paper provides an overview of geometric and the related geometric-like processes (GLP). We include a brief review of the geometric process and some basic definitions, facts and references for geometric-like processes. Some of their applications to areas such as maintenance, reliability, warranty analysis and others, along with appropriate references, are also outlined. Finally, we list a number of open research questions related to GLP.

Suggested Citation

  • Arnold, Richard & Chukova, Stefanka & Hayakawa, Yu & Marshall, Sarah, 2020. "Geometric-Like Processes: An Overview and Some Reliability Applications," Reliability Engineering and System Safety, Elsevier, vol. 201(C).
    • Handle: RePEc:eee:reensy:v:201:y:2020:i:c:s0951832020304919
    DOI: 10.1016/j.ress.2020.106990
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0951832020304919
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ress.2020.106990?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Richard Arnold & Stefanka Chukova & Yu Hayakawa & Sarah Marshall, 2019. "Warranty cost analysis with an alternating geometric process," Journal of Risk and Reliability, , vol. 233(4), pages 698-715, August.
    2. Wan, Wai-Yin & Chan, Jennifer So-Kuen, 2011. "Bayesian analysis of robust Poisson geometric process model using heavy-tailed distributions," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 687-702, January.
    3. Jennifer Chan & Doris Leung, 2010. "Binary geometric process model for the modeling of longitudinal binary data with trend," Computational Statistics, Springer, vol. 25(3), pages 505-536, September.
    4. Wang, Guan Jun & Zhang, Yuan Lin, 2013. "Optimal repair–replacement policies for a system with two types of failures," European Journal of Operational Research, Elsevier, vol. 226(3), pages 500-506.
    5. Chen, Jinyuan & Li, Zehui, 2008. "An extended extreme shock maintenance model for a deteriorating system," Reliability Engineering and System Safety, Elsevier, vol. 93(8), pages 1123-1129.
    6. Lam Yeh & So Kuen Chan, 1998. "Statistical inference for geometric processes with lognormal distribution," Computational Statistics & Data Analysis, Elsevier, vol. 27(1), pages 99-112, March.
    7. Zhang, Yuan Lin & Wang, Guan Jun, 2009. "A geometric process repair model for a repairable cold standby system with priority in use and repair," Reliability Engineering and System Safety, Elsevier, vol. 94(11), pages 1782-1787.
    8. Sarah Marshall & Richard Arnold & Stefanka Chukova & Yu Hayakawa, 2018. "Warranty cost analysis: Increasing warranty repair times," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 34(4), pages 544-561, July.
    9. Leung, Kit Nam Francis & Zhang, Yuan Lin & Lai, Kin Keung, 2011. "Analysis for a two-dissimilar-component cold standby repairable system with repair priority," Reliability Engineering and System Safety, Elsevier, vol. 96(11), pages 1542-1551.
    10. J.S.K. Chan & W.Y. Wan & P.L.H. Yu, 2014. "A Poisson geometric process approach for predicting drop-out and committed first-time blood donors," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(7), pages 1486-1503, July.
    11. Chen, Jianwei & Li, Kim-Hung & Lam, Yeh, 2010. "Bayesian computation for geometric process in maintenance problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(4), pages 771-781.
    12. Lam, Yeh & Zhang, Yuan Lin & Zheng, Yao Hui, 2002. "A geometric process equivalent model for a multistate degenerative system," European Journal of Operational Research, Elsevier, vol. 142(1), pages 21-29, October.
    13. Aydoğdu, Halil & Karabulut, İhsan & Şen, Elif, 2013. "On the exact distribution and mean value function of a geometric process with exponential interarrival times," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2577-2582.
    14. Yuan-Lin Zhang & Guan-Jun Wang, 2017. "An extended geometric process repair model with delayed repair and slight failure type," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(1), pages 427-437, January.
    15. Lam, Yeh, 2007. "A geometric process maintenance model with preventive repair," European Journal of Operational Research, Elsevier, vol. 182(2), pages 806-819, October.
    16. Zhang, Yuan Lin, 2007. "A discussion on "A bivariate optimal replacement policy for a repairable system"," European Journal of Operational Research, Elsevier, vol. 179(1), pages 275-276, May.
    17. G.Q. Cheng & L. Li, 2012. "A geometric process repair model with inspections and its optimisation," International Journal of Systems Science, Taylor & Francis Journals, vol. 43(9), pages 1650-1655.
    18. Chan, Jennifer S. K. & Lam, Yeh & Leung, Doris Y. P., 2004. "Statistical inference for geometric processes with gamma distributions," Computational Statistics & Data Analysis, Elsevier, vol. 47(3), pages 565-581, October.
    19. Castro, I.T. & Pérez-Ocón, R., 2006. "Reward optimization of a repairable system," Reliability Engineering and System Safety, Elsevier, vol. 91(3), pages 311-319.
    20. Xiaolin Liang & Yeh Lam & Zehui Li, 2011. "Optimal replacement policy for a general geometric process model with δ-shock," International Journal of Systems Science, Taylor & Francis Journals, vol. 42(12), pages 2021-2034.
    21. Shaomin Wu, 2018. "Doubly geometric processes and applications," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 69(1), pages 66-77, January.
    22. Yu, Miaomiao & Tang, Yinghui & Liu, Liping & Cheng, Jiang, 2013. "A phase-type geometric process repair model with spare device procurement and repairman’s multiple vacations," European Journal of Operational Research, Elsevier, vol. 225(2), pages 310-323.
    23. Y. L. Zhang & G. J. Wang, 2018. "An extended geometric process repair model with imperfect delayed repair under different objective functions," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(13), pages 3204-3219, July.
    24. Stefanka Chukova & Yu Hayakawa, 2004. "Warranty cost analysis: non‐zero repair time," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 20(1), pages 59-71, January.
    25. Aydogdu, Halil & Kara, Mahmut, 2012. "Nonparametric estimation in [alpha]-series processes," Computational Statistics & Data Analysis, Elsevier, vol. 56(1), pages 190-201, January.
    26. W. John Braun & Wei Li & Yiqiang Q. Zhao, 2005. "Properties of the geometric and related processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(7), pages 607-616, October.
    27. Miaomiao Yu & Yinghui Tang & Yonghong Fu, 2013. "A geometric process model for /(/)/1/ queue with new service machine procurement lead time," International Journal of Systems Science, Taylor & Francis Journals, vol. 44(6), pages 1061-1075.
    28. Lam, Yeh & Zhang, Yuan Lin & Liu, Qun, 2006. "A geometric process model for M/M/1 queueing system with a repairable service station," European Journal of Operational Research, Elsevier, vol. 168(1), pages 100-121, January.
    29. Chan, Jennifer So Kuen & Wan, Wai Yin, 2014. "Multivariate generalized Poisson geometric process model with scale mixtures of normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 72-87.
    30. Yuan, Li & Xu, Jian, 2011. "An optimal replacement policy for a repairable system based on its repairman having vacations," Reliability Engineering and System Safety, Elsevier, vol. 96(7), pages 868-875.
    31. Chan, J.S.K. & Lam, C.P.Y. & Yu, P.L.H. & Choy, S.T.B. & Chen, C.W.S., 2012. "A Bayesian conditional autoregressive geometric process model for range data," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3006-3019.
    32. Zhang, Yuan Lin & Yam, Richard C.M. & Zuo, Ming J., 2007. "A bivariate optimal replacement policy for a multistate repairable system," Reliability Engineering and System Safety, Elsevier, vol. 92(4), pages 535-542.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hu, Wei & Westerlund, Per & Hilber, Patrik & Chen, Chuanhai & Yang, Zhaojun, 2022. "A general model, estimation, and procedure for modeling recurrent failure process of high-voltage circuit breakers considering multivariate impacts," Reliability Engineering and System Safety, Elsevier, vol. 220(C).
    2. An, Youjun & Chen, Xiaohui & Hu, Jiawen & Zhang, Lin & Li, Yinghe & Jiang, Junwei, 2022. "Joint optimization of preventive maintenance and production rescheduling with new machine insertion and processing speed selection," Reliability Engineering and System Safety, Elsevier, vol. 220(C).
    3. Rasay, Hasan & Taghipour, Sharareh & Sharifi, Mani, 2022. "An integrated Maintenance and Statistical Process Control Model for a Deteriorating Production Process," Reliability Engineering and System Safety, Elsevier, vol. 228(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Guan Jun Wang & Yuan Lin Zhang, 2016. "Optimal replacement policy for a two-dissimilar-component cold standby system with different repair actions," International Journal of Systems Science, Taylor & Francis Journals, vol. 47(5), pages 1021-1031, April.
    2. Chan, Jennifer So Kuen & Wan, Wai Yin, 2014. "Multivariate generalized Poisson geometric process model with scale mixtures of normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 72-87.
    3. Yu, Miaomiao & Tang, Yinghui & Liu, Liping & Cheng, Jiang, 2013. "A phase-type geometric process repair model with spare device procurement and repairman’s multiple vacations," European Journal of Operational Research, Elsevier, vol. 225(2), pages 310-323.
    4. Miaomiao Yu & Yinghui Tang, 2024. "Analyze periodic inspection and replacement policy of a shock and wear model with phase-type inter-shock arrival times using roots method," Journal of Risk and Reliability, , vol. 238(2), pages 233-246, April.
    5. Leung, Kit Nam Francis & Zhang, Yuan Lin & Lai, Kin Keung, 2011. "Analysis for a two-dissimilar-component cold standby repairable system with repair priority," Reliability Engineering and System Safety, Elsevier, vol. 96(11), pages 1542-1551.
    6. Zhang, Yuan Lin & Wang, Guan Jun, 2009. "A geometric process repair model for a repairable cold standby system with priority in use and repair," Reliability Engineering and System Safety, Elsevier, vol. 94(11), pages 1782-1787.
    7. Mingjuan Sun & Qinglai Dong & Zihan Gao, 2022. "An Imperfect Repair Model with Delayed Repair under Replacement and Repair Thresholds," Mathematics, MDPI, vol. 10(13), pages 1-15, June.
    8. Wenke Gao, 2020. "An extended geometric process and its application in replacement policy," Journal of Risk and Reliability, , vol. 234(1), pages 88-103, February.
    9. Sarada, Y. & Shenbagam, R., 2021. "Optimization of a repairable deteriorating system subject to random threshold failure using preventive repair and stochastic lead time," Reliability Engineering and System Safety, Elsevier, vol. 205(C).
    10. Junyuan Wang & Jimin Ye, 2022. "A new repair model and its optimization for cold standby system," Operational Research, Springer, vol. 22(1), pages 105-122, March.
    11. Miaomiao Yu & Yinghui Tang, 2017. "Optimal replacement policy based on maximum repair time for a random shock and wear model," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 80-94, April.
    12. J.S.K. Chan & W.Y. Wan & P.L.H. Yu, 2014. "A Poisson geometric process approach for predicting drop-out and committed first-time blood donors," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(7), pages 1486-1503, July.
    13. Liu, Baoliang & Cui, Lirong & Wen, Yanqing & Shen, Jingyuan, 2015. "A cold standby repairable system with working vacations and vacation interruption following Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 1-8.
    14. Delia Montoro-Cazorla & Rafael Pérez-Ocón, 2022. "Optimizing Costs in a Reliability System under Markovian Arrival of Failures and Reposition by K -Policy Inspection," Mathematics, MDPI, vol. 10(11), pages 1-21, June.
    15. Jennifer Chan & Doris Leung, 2010. "Binary geometric process model for the modeling of longitudinal binary data with trend," Computational Statistics, Springer, vol. 25(3), pages 505-536, September.
    16. Junyuan Wang & Jimin Ye & Qianru Ma & Pengfei Xie, 2022. "An extended geometric process repairable model with its repairman having vacation," Annals of Operations Research, Springer, vol. 311(1), pages 401-415, April.
    17. Yevkin, Alexander & Krivtsov, Vasiliy, 2020. "A generalized model for recurrent failures prediction," Reliability Engineering and System Safety, Elsevier, vol. 204(C).
    18. Yang, Dong-Yuh & Tsao, Chih-Lung, 2019. "Reliability and availability analysis of standby systems with working vacations and retrial of failed components," Reliability Engineering and System Safety, Elsevier, vol. 182(C), pages 46-55.
    19. Richard Arnold & Stefanka Chukova & Yu Hayakawa & Sarah Marshall, 2019. "Warranty cost analysis with an alternating geometric process," Journal of Risk and Reliability, , vol. 233(4), pages 698-715, August.
    20. An, Youjun & Chen, Xiaohui & Hu, Jiawen & Zhang, Lin & Li, Yinghe & Jiang, Junwei, 2022. "Joint optimization of preventive maintenance and production rescheduling with new machine insertion and processing speed selection," Reliability Engineering and System Safety, Elsevier, vol. 220(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:reensy:v:201:y:2020:i:c:s0951832020304919. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/reliability-engineering-and-system-safety .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.