{"id":"https://openalex.org/W2044456301","doi":"https://doi.org/10.1109/tr.2011.2182399","title":"An Approximate Solution to the G\u2013Renewal Equation With an Underlying Weibull Distribution","display_name":"An Approximate Solution to the G\u2013Renewal Equation With an Underlying Weibull Distribution","publication_year":2012,"publication_date":"2012-02-01","ids":{"openalex":"https://openalex.org/W2044456301","doi":"https://doi.org/10.1109/tr.2011.2182399","mag":"2044456301"},"language":"en","primary_location":{"id":"doi:10.1109/tr.2011.2182399","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tr.2011.2182399","pdf_url":null,"source":{"id":"https://openalex.org/S87725633","display_name":"IEEE Transactions on Reliability","issn_l":"0018-9529","issn":["0018-9529","1558-1721"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Reliability","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026130406","display_name":"Olexandr Yevkin","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Olexandr Yevkin","raw_affiliation_strings":["Dyadem International Limited, Toronto, ONT, Canada","Dyaden Int. Ltd., Toronto, ON, Canada"],"affiliations":[{"raw_affiliation_string":"Dyadem International Limited, Toronto, ONT, Canada","institution_ids":[]},{"raw_affiliation_string":"Dyaden Int. Ltd., Toronto, ON, Canada","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5045299654","display_name":"Vasiliy Krivtsov","orcid":"https://orcid.org/0000-0003-1351-6180"},"institutions":[{"id":"https://openalex.org/I1292974536","display_name":"Ford Motor Company (United States)","ror":"https://ror.org/00g2tkw06","country_code":"US","type":"company","lineage":["https://openalex.org/I1292974536"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Vasiliy Krivtsov","raw_affiliation_strings":["Ford Motor Company Limited, Dearborn, USA","Ford Motor Co., Dearborn, MI, , USA"],"affiliations":[{"raw_affiliation_string":"Ford Motor Company Limited, Dearborn, USA","institution_ids":["https://openalex.org/I1292974536"]},{"raw_affiliation_string":"Ford Motor Co., Dearborn, MI, , USA","institution_ids":["https://openalex.org/I1292974536"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5026130406"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.4534,"has_fulltext":false,"cited_by_count":21,"citation_normalized_percentile":{"value":0.88549773,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"61","issue":"1","first_page":"68","last_page":"73"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10780","display_name":"Reliability and Maintenance Optimization","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/2213","display_name":"Safety, Risk, Reliability and Quality"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/weibull-distribution","display_name":"Weibull distribution","score":0.8332406282424927},{"id":"https://openalex.org/keywords/renewal-theory","display_name":"Renewal theory","score":0.8058798313140869},{"id":"https://openalex.org/keywords/monte-carlo-method","display_name":"Monte Carlo method","score":0.6899942755699158},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5540215969085693},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5372740030288696},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.48732075095176697},{"id":"https://openalex.org/keywords/exponential-distribution","display_name":"Exponential distribution","score":0.4319410026073456},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4284582734107971},{"id":"https://openalex.org/keywords/process","display_name":"Process (computing)","score":0.42004936933517456},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3751384913921356},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.21547454595565796}],"concepts":[{"id":"https://openalex.org/C173291955","wikidata":"https://www.wikidata.org/wiki/Q732332","display_name":"Weibull distribution","level":2,"score":0.8332406282424927},{"id":"https://openalex.org/C110507122","wikidata":"https://www.wikidata.org/wiki/Q1356856","display_name":"Renewal theory","level":2,"score":0.8058798313140869},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.6899942755699158},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5540215969085693},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5372740030288696},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.48732075095176697},{"id":"https://openalex.org/C55350006","wikidata":"https://www.wikidata.org/wiki/Q237193","display_name":"Exponential distribution","level":2,"score":0.4319410026073456},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4284582734107971},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.42004936933517456},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3751384913921356},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.21547454595565796},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tr.2011.2182399","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tr.2011.2182399","pdf_url":null,"source":{"id":"https://openalex.org/S87725633","display_name":"IEEE Transactions on Reliability","issn_l":"0018-9529","issn":["0018-9529","1558-1721"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Reliability","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W239846830","https://openalex.org/W1533220057","https://openalex.org/W1962447254","https://openalex.org/W1969543203","https://openalex.org/W1979912268","https://openalex.org/W1990064670","https://openalex.org/W1997255375","https://openalex.org/W2061625717","https://openalex.org/W2062898928","https://openalex.org/W2077346801","https://openalex.org/W2088123622","https://openalex.org/W2107305576","https://openalex.org/W2118931891","https://openalex.org/W2127332440","https://openalex.org/W2134270577","https://openalex.org/W2141923571","https://openalex.org/W2333379900","https://openalex.org/W2490166567","https://openalex.org/W2738732597","https://openalex.org/W4212904528"],"related_works":["https://openalex.org/W1620508096","https://openalex.org/W2357734103","https://openalex.org/W2094562318","https://openalex.org/W1490217699","https://openalex.org/W3195534432","https://openalex.org/W2012284326","https://openalex.org/W3165414413","https://openalex.org/W4242280303","https://openalex.org/W1996480520","https://openalex.org/W2127782827"],"abstract_inverted_index":{"An":[0],"important":[1],"characteristic":[2],"of":[3,7,21,27,87,95,116],"the":[4,12,17,36,40,51,79,88,93,100,107,117],"g-renewal":[5,13,41,118],"process,":[6,35],"great":[8],"practical":[9],"interest,":[10],"is":[11,38,64,83,91],"equation,":[14],"which":[15],"represents":[16],"expected":[18],"cumulative":[19],"number":[20],"recurrent":[22],"events":[23],"as":[24],"a":[25,46,70],"function":[26],"time.":[28],"Just":[29],"like":[30],"in":[31,92],"an":[32,73,113],"ordinary":[33],"renewal":[34],"problem":[37],"that":[39],"equation":[42],"does":[43],"not":[44],"have":[45],"closed":[47],"form":[48],"solution,":[49,61,77,109],"unless":[50],"underlying":[52,80],"event":[53],"times":[54],"are":[55],"exponentially":[56],"distributed.":[57],"The":[58,85],"Monte":[59,102],"Carlo":[60,103],"although":[62],"exhaustive,":[63],"computationally":[65],"demanding.":[66],"This":[67],"paper":[68],"offers":[69],"simple-to-implement":[71],"(in":[72],"Excel":[74],"spreadsheet)":[75],"approximate":[76],"when":[78,97],"failure-time":[81],"distribution":[82],"Weibull.":[84],"accuracy":[86],"proposed":[89,108],"solution":[90],"neighborhood":[94],"2%,":[96],"compared":[98],"to":[99],"respective":[101],"solution.":[104],"Based":[105],"on":[106],"we":[110],"also":[111],"consider":[112],"estimation":[114],"procedure":[115],"process":[119],"parameters.":[120]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2013,"cited_by_count":5}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
