{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,11]],"date-time":"2026-01-11T02:04:46Z","timestamp":1768097086297,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T00:00:00Z","timestamp":1672876800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Autonomous University of Ciudad Ju\u00e1rez"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Degradation modeling requires to consider the complexity of both the internal structure of highly reliable products and the environmental conditions, to define appropriate models to obtain estimations about the reliability and quality. These conditions reflect different sources of variability that need to be considered in the aims of obtaining accurate estimations. Although several models have been proposed in the literature, few of them consider several simultaneous sources of variability. In this paper, we propose a model based on the gamma process that considers three sources of variability, specifically in the threshold, the initial level of degradation, and in the scale parameter of the gamma process. The model considers a convolution operation of the threshold and the initial level to then be characterized via numerical integration with the gamma process with random scale. The obtained results showed that the model can be used to model the degradation of products with these sources of variability, which means that it can used for case studies where both the initial level and threshold are inherently random and the randomness in the scale parameter can be proved. The performance is illustrated with a comprehensive simulation study and with the application in a case study.<\/jats:p>","DOI":"10.3390\/sym15010162","type":"journal-article","created":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T01:48:19Z","timestamp":1672969699000},"page":"162","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Gamma Process with Three Sources of Variability"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2951-2344","authenticated-orcid":false,"given":"Luis Alberto","family":"Rodr\u00edguez-Pic\u00f3n","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering and Manufacturing, Autonomous University of Ciudad Ju\u00e1rez, Av. Plutarco El\u00edas Calles 1210, Fovissste Chamizal, Ciudad Ju\u00e1rez 32310, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2533-0036","authenticated-orcid":false,"given":"Luis Carlos","family":"M\u00e9ndez-Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering and Manufacturing, Autonomous University of Ciudad Ju\u00e1rez, Av. Plutarco El\u00edas Calles 1210, Fovissste Chamizal, Ciudad Ju\u00e1rez 32310, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2445-0500","authenticated-orcid":false,"given":"Iv\u00e1n Juan Carlos","family":"P\u00e9rez-Olgu\u00edn","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering and Manufacturing, Autonomous University of Ciudad Ju\u00e1rez, Av. Plutarco El\u00edas Calles 1210, Fovissste Chamizal, Ciudad Ju\u00e1rez 32310, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3807-565X","authenticated-orcid":false,"given":"Jes\u00fas Israel","family":"Hern\u00e1ndez-Hern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Autonomous University of Ciudad Ju\u00e1rez, Av. Plutarco El\u00edas Calles 1210, Fovissste Chamizal, Ciudad Ju\u00e1rez 32310, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.ress.2007.03.019","article-title":"A survey of the application of gamma processes in maintenance","volume":"94","year":"2009","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/j.solener.2017.03.061","article-title":"Application of gamma process model to estimate the lifetime of photovoltaic modules","volume":"147","author":"Park","year":"2017","journal-title":"Sol. Energy"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107912","DOI":"10.1016\/j.ress.2021.107912","article-title":"Estimation of the value of an inspection and maintenance program: A Bayesian gamma process model","volume":"216","author":"Yuan","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.strusafe.2013.09.001","article-title":"Gamma degradation models for earthquake-resistant structures","volume":"45","author":"Iervolino","year":"2013","journal-title":"Struct. Saf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"106496","DOI":"10.1016\/j.ress.2019.106496","article-title":"Gamma process based optimal mission abort policy","volume":"190","author":"Qiu","year":"2019","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"10581","DOI":"10.1109\/ACCESS.2018.2799853","article-title":"Gamma degradation process and accelerated model combined reliability analysis method for rubber O-rings","volume":"6","author":"Sun","year":"2018","journal-title":"IEEE Access"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.microrel.2015.12.006","article-title":"Lifetime estimation of LED lamp using gamma process model","volume":"57","author":"Park","year":"2016","journal-title":"Microelectron. Reliab."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/JPHOT.2019.2950472","article-title":"Lumen degradation lifetime prediction for high-power white LEDs based on the gamma process model","volume":"11","author":"Ibrahim","year":"2019","journal-title":"IEEE Photon. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"107090","DOI":"10.1016\/j.polymertesting.2021.107090","article-title":"A Gamma process-based degradation testing of silicone encapsulant used in LED packaging","volume":"96","author":"Fan","year":"2021","journal-title":"Polym. Test."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107797","DOI":"10.1016\/j.ress.2021.107797","article-title":"Prognostics for lithium-ion batteries using a two-phase gamma degradation process model","volume":"214","author":"Lin","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hu, C., Fan, H., and Wang, Z. (2021). Gamma process-based degradation modeling and residual life prediction. Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, Springer.","DOI":"10.1007\/978-981-16-2267-0"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"012001","DOI":"10.1088\/1742-6596\/1820\/1\/012001","article-title":"Reliability analysis of wind turbine based on degradation threshold","volume":"1820","author":"Mo","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zhang, L., Xu, A., An, L., and Li, M. (2022). Bayesian inference of system reliability for multicomponent stress-strength model under Marshall-Olkin Weibull distribution. Systems, 10.","DOI":"10.3390\/systems10060196"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.2478\/mme-2018-0101","article-title":"Simulation of bearing degradation by the use of the gamma stochastic process","volume":"22","author":"Bouzaouit","year":"2018","journal-title":"Mechan. Mechan. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/j.ress.2015.05.011","article-title":"Reliability demonstration methodology for products with gamma process by optimal accelerated degradation testing","volume":"142","author":"Zhang","year":"2015","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1023\/B:LIDA.0000036389.14073.dd","article-title":"Covariates and random effects in a gamma process model with application to degradation and failure","volume":"10","author":"Lawless","year":"2004","journal-title":"Lifetime Data Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1200","DOI":"10.1016\/j.ejor.2020.11.036","article-title":"Degradation data analysis based on gamma process with random effects","volume":"292","author":"Wang","year":"2021","journal-title":"Eur. J. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Liu, G., Guan, Q., Tang, Y., and Tzeng, Y. (2022). Interval modeling for gamma process degradation model. Symmetry, 14.","DOI":"10.3390\/sym14050954"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.cja.2014.12.015","article-title":"Lifetime prediction based on gamma processes from accelerated degradation data","volume":"28","author":"Wang","year":"2015","journal-title":"Chin. J. Aeronaut."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.ress.2018.06.019","article-title":"Bayesian model averaging based reliability analysis method for monotonic degradation dataset based on inverse Gaussian process and gamma process","volume":"180","author":"Liu","year":"2018","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"107504","DOI":"10.1016\/j.ress.2021.107504","article-title":"Remaining useful life prediction and optimal maintenance time determination for a single unit using isotonic regression and gamma process model","volume":"210","author":"Wang","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.engfracmech.2015.05.027","article-title":"A gamma process model for the analysis of fatigue crack growth data","volume":"142","author":"Guida","year":"2015","journal-title":"Eng. Fract. Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1002\/qre.2271","article-title":"The transformed gamma process for degradation phenomena in presence of unexplained forms of unit-to-unit variability","volume":"34","author":"Giorgio","year":"2018","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Rodr\u00edguez-Pic\u00f3n, L.A., Rodr\u00edguez-Pic\u00f3n, A.P., M\u00e9ndez-Gonz\u00e1lez, L.C., Rodr\u00edguez-Borb\u00f3n, M.I., and Alvarado-Iniesta, A. (2018). Degradation modeling based on gamma process models with random effects. Communications in Statistics\u2014Simulation and Computation, Taylor & Francis.","DOI":"10.1080\/03610918.2017.1324981"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"243648","DOI":"10.1155\/2015\/243648","article-title":"LED lighting system reliability modeling and inference via random effects gamma process and copula function","volume":"2015","author":"Hao","year":"2015","journal-title":"Int. J. Photoenergy"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8","DOI":"10.17531\/ein.2017.1.2","article-title":"Reliability assessment for systems with two performance characteristics based on gamma processes with marginal heterogeneous random effects","volume":"19","year":"2016","journal-title":"Eksploat. Niezawodn. Maint. Reliab."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/j.cie.2018.09.003","article-title":"Planning of step-stress accelerated degradation test based on non-stationary gamma process with random effects","volume":"125","author":"Duan","year":"2018","journal-title":"Comput. Ind. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1109\/TR.2012.2194351","article-title":"Optimal design for degradation tests based on gamma processes with random effects","volume":"61","author":"Tsai","year":"2012","journal-title":"IEEE Trans. Reliab."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1080\/00401706.2013.869261","article-title":"Semiparametric estimation of gamma processes for deteriorating products","volume":"56","author":"Ye","year":"2014","journal-title":"Technometrics"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"108136","DOI":"10.1016\/j.ress.2021.108136","article-title":"Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints","volume":"218","author":"Luo","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Rodr\u00edguez-Pic\u00f3n, L.A., Mendez-Gonzalez, L.C., Flores-Ochoa, V.H., Perez-Olguin, I.J.C., and Garcia, V. (2022). Degradation modeling based on the gamma process with random initial degradation level and random threshold. Qual. Technol. Quant. Manag., in press.","DOI":"10.1080\/16843703.2022.2146904"},{"key":"ref_32","unstructured":"Goulet, V. (2022, October 30). Expint: Exponential Integral and Incomplete Gamma Function. R Package. Available online: https:\/\/cran.r-project.org\/package=expint."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/162\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:00:40Z","timestamp":1760119240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/162"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,5]]},"references-count":32,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010162"],"URL":"https:\/\/doi.org\/10.3390\/sym15010162","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,5]]}}}