{"id":"https://openalex.org/W4412379966","doi":"https://doi.org/10.3389/fmtec.2025.1439421","title":"Extraction of exact symbolic stationary probability formulas for Markov chains with finite space with application to production lines. Part I: description of methodology","display_name":"Extraction of exact symbolic stationary probability formulas for Markov chains with finite space with application to production lines. Part I: description of methodology","publication_year":2025,"publication_date":"2025-07-14","ids":{"openalex":"https://openalex.org/W4412379966","doi":"https://doi.org/10.3389/fmtec.2025.1439421"},"language":"en","primary_location":{"id":"doi:10.3389/fmtec.2025.1439421","doi":"https://doi.org/10.3389/fmtec.2025.1439421","is_oa":true,"landing_page_url":"https://doi.org/10.3389/fmtec.2025.1439421","pdf_url":"https://www.frontiersin.org/journals/manufacturing-technology/articles/10.3389/fmtec.2025.1439421/pdf","source":{"id":"https://openalex.org/S4387279249","display_name":"Frontiers in Manufacturing Technology","issn_l":"2813-0359","issn":["2813-0359"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1"],"host_organization":"https://openalex.org/P4310320527","host_organization_name":"Frontiers Media","host_organization_lineage":["https://openalex.org/P4310320527"],"host_organization_lineage_names":["Frontiers Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Frontiers in Manufacturing Technology","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.frontiersin.org/journals/manufacturing-technology/articles/10.3389/fmtec.2025.1439421/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5075852625","display_name":"Konstantinos S. Boulas","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I98805295","display_name":"University of the Aegean","ror":"https://ror.org/03zsp3p94","country_code":"GR","type":"education","lineage":["https://openalex.org/I98805295"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Konstantinos S. Boulas","raw_affiliation_strings":["Management and Decision Engineering Laboratory (MDE-Lab), Department of Financial and Management Engineering, School of Engineering, University of the Aegean, Chios, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Management and Decision Engineering Laboratory (MDE-Lab), Department of Financial and Management Engineering, School of Engineering, University of the Aegean, Chios, Greece","institution_ids":["https://openalex.org/I98805295"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5112247169","display_name":"Georgios D. Dounias","orcid":"https://orcid.org/0000-0002-8130-1766","observed_orcids":["https://orcid.org/0000-0002-8130-1766"]},"institutions":[{"id":"https://openalex.org/I98805295","display_name":"University of the Aegean","ror":"https://ror.org/03zsp3p94","country_code":"GR","type":"education","lineage":["https://openalex.org/I98805295"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Georgios D. Dounias","raw_affiliation_strings":["Management and Decision Engineering Laboratory (MDE-Lab), Department of Financial and Management Engineering, School of Engineering, University of the Aegean, Chios, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Management and Decision Engineering Laboratory (MDE-Lab), Department of Financial and Management Engineering, School of Engineering, University of the Aegean, Chios, Greece","institution_ids":["https://openalex.org/I98805295"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079148533","display_name":"Chrissoleon T. Papadopoulos","orcid":"https://orcid.org/0000-0002-6721-5175","observed_orcids":["https://orcid.org/0000-0002-6721-5175"]},"institutions":[{"id":"https://openalex.org/I21370196","display_name":"Aristotle University of Thessaloniki","ror":"https://ror.org/02j61yw88","country_code":"GR","type":"education","lineage":["https://openalex.org/I21370196"]}],"countries":["GR"],"is_corresponding":true,"raw_author_name":"Chrissoleon T. Papadopoulos","raw_affiliation_strings":["School of Economic Sciences, Aristotle University of Thessaloniki, Thessaloniki, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Economic Sciences, Aristotle University of Thessaloniki, Thessaloniki, Greece","institution_ids":["https://openalex.org/I21370196"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5079148533"],"corresponding_institution_ids":["https://openalex.org/I21370196"],"apc_list":{"value":1950,"currency":"CHF","value_usd":2353},"apc_paid":{"value":1950,"currency":"CHF","value_usd":2353},"fwci":1.4029,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.84143781,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"5","issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10974","display_name":"Advanced Queuing Theory Analysis","score":0.4375,"subfield":{"id":"https://openalex.org/subfields/1404","display_name":"Management Information Systems"},"field":{"id":"https://openalex.org/fields/14","display_name":"Business, Management and Accounting"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10974","display_name":"Advanced Queuing Theory Analysis","score":0.4375,"subfield":{"id":"https://openalex.org/subfields/1404","display_name":"Management Information Systems"},"field":{"id":"https://openalex.org/fields/14","display_name":"Business, Management and Accounting"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.3190999925136566,"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/T11125","display_name":"Petri Nets in System Modeling","score":0.07100000232458115,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/markov-chains","display_name":"Markov chains","score":0.9549999833106995},{"id":"https://openalex.org/keywords/stationary-distribution","display_name":"stationary distribution","score":0.8980000019073486},{"id":"https://openalex.org/keywords/symbolic-computation","display_name":"symbolic computation","score":0.5139999985694885},{"id":"https://openalex.org/keywords/analytical-solution","display_name":"analytical solution","score":0.6930000185966492},{"id":"https://openalex.org/keywords/production-line","display_name":"production line","score":0.6729999780654907},{"id":"https://openalex.org/keywords/spanning-trees","display_name":"spanning trees","score":0.7760000228881836},{"id":"https://openalex.org/keywords/transition-matrix","display_name":"transition matrix","score":0.546999990940094},{"id":"https://openalex.org/keywords/state-transition-diagrams","display_name":"state diagrams","score":0.6930000185966492},{"id":"https://openalex.org/keywords/depth-first-search","display_name":"depth-first search","score":0.9139999747276306},{"id":"https://openalex.org/keywords/performance-indicators","display_name":"performance indicators","score":0.6779999732971191},{"id":"https://openalex.org/keywords/queueing-theory","display_name":"queueing theory","score":0.722000002861023}],"concepts":[{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.713365912437439},{"id":"https://openalex.org/C2778348673","wikidata":"https://www.wikidata.org/wiki/Q739302","display_name":"Production (economics)","level":2,"score":0.559220016002655},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5278037786483765},{"id":"https://openalex.org/C4725764","wikidata":"https://www.wikidata.org/wiki/Q844704","display_name":"Extraction (chemistry)","level":2,"score":0.4793297052383423},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4681404232978821},{"id":"https://openalex.org/C76408418","wikidata":"https://www.wikidata.org/wiki/Q5419314","display_name":"Examples of Markov chains","level":5,"score":0.4633849561214447},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42208585143089294},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.4162988066673279},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38326749205589294},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.36463767290115356},{"id":"https://openalex.org/C163836022","wikidata":"https://www.wikidata.org/wiki/Q6771326","display_name":"Markov model","level":3,"score":0.31394559144973755},{"id":"https://openalex.org/C54907487","wikidata":"https://www.wikidata.org/wiki/Q7915688","display_name":"Variable-order Markov model","level":4,"score":0.24274322390556335},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.23218652606010437},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10690310597419739},{"id":"https://openalex.org/C43617362","wikidata":"https://www.wikidata.org/wiki/Q170050","display_name":"Chromatography","level":1,"score":0.06611171364784241},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.05979052186012268},{"id":"https://openalex.org/C139719470","wikidata":"https://www.wikidata.org/wiki/Q39680","display_name":"Macroeconomics","level":1,"score":0.0},{"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3389/fmtec.2025.1439421","doi":"https://doi.org/10.3389/fmtec.2025.1439421","is_oa":true,"landing_page_url":"https://doi.org/10.3389/fmtec.2025.1439421","pdf_url":"https://www.frontiersin.org/journals/manufacturing-technology/articles/10.3389/fmtec.2025.1439421/pdf","source":{"id":"https://openalex.org/S4387279249","display_name":"Frontiers in Manufacturing Technology","issn_l":"2813-0359","issn":["2813-0359"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1"],"host_organization":"https://openalex.org/P4310320527","host_organization_name":"Frontiers Media","host_organization_lineage":["https://openalex.org/P4310320527"],"host_organization_lineage_names":["Frontiers Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Frontiers in Manufacturing Technology","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:55018f30ec204fbf9e4fda75b6e962de","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/55018f30ec204fbf9e4fda75b6e962de","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Frontiers in Manufacturing Technology, Vol 5 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3389/fmtec.2025.1439421","doi":"https://doi.org/10.3389/fmtec.2025.1439421","is_oa":true,"landing_page_url":"https://doi.org/10.3389/fmtec.2025.1439421","pdf_url":"https://www.frontiersin.org/journals/manufacturing-technology/articles/10.3389/fmtec.2025.1439421/pdf","source":{"id":"https://openalex.org/S4387279249","display_name":"Frontiers in Manufacturing Technology","issn_l":"2813-0359","issn":["2813-0359"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1"],"host_organization":"https://openalex.org/P4310320527","host_organization_name":"Frontiers Media","host_organization_lineage":["https://openalex.org/P4310320527"],"host_organization_lineage_names":["Frontiers Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Frontiers in Manufacturing Technology","raw_type":"journal-article"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4412379966.pdf","grobid_xml":"https://content.openalex.org/works/W4412379966.grobid-xml"},"referenced_works_count":40,"referenced_works":["https://openalex.org/W9167880","https://openalex.org/W121824094","https://openalex.org/W997137459","https://openalex.org/W1556096280","https://openalex.org/W1576818901","https://openalex.org/W1985468884","https://openalex.org/W1986646858","https://openalex.org/W1995875735","https://openalex.org/W2007949375","https://openalex.org/W2022871748","https://openalex.org/W2041709301","https://openalex.org/W2052256872","https://openalex.org/W2053392780","https://openalex.org/W2055336889","https://openalex.org/W2058120853","https://openalex.org/W2075721872","https://openalex.org/W2076406842","https://openalex.org/W2110099831","https://openalex.org/W2134676378","https://openalex.org/W2138002711","https://openalex.org/W2330005588","https://openalex.org/W2343061998","https://openalex.org/W2398983037","https://openalex.org/W2563927543","https://openalex.org/W2745324256","https://openalex.org/W2887060483","https://openalex.org/W2905022744","https://openalex.org/W2906547263","https://openalex.org/W2923745563","https://openalex.org/W3037315640","https://openalex.org/W3124556755","https://openalex.org/W3128031213","https://openalex.org/W3208002035","https://openalex.org/W4212971671","https://openalex.org/W4233121052","https://openalex.org/W4236609033","https://openalex.org/W4244035502","https://openalex.org/W4293179730","https://openalex.org/W4302604684","https://openalex.org/W4412380163"],"related_works":["https://openalex.org/W2379651310","https://openalex.org/W23237351","https://openalex.org/W3023326395","https://openalex.org/W2987631717","https://openalex.org/W190459708","https://openalex.org/W50519898","https://openalex.org/W1541249122","https://openalex.org/W2077211377","https://openalex.org/W2075517632","https://openalex.org/W2989366387"],"abstract_inverted_index":{"Introduction":[0],"Markov":[1,38,53,80,118,153,248,279,319],"chains":[2,54,320],"are":[3,19,55,292],"a":[4,28,33,79,177,184,196,259,274,278],"powerful":[5],"tool":[6],"for":[7,199,224,313],"modeling":[8],"systems":[9],"in":[10,113,218,304,331],"various":[11],"scientific":[12],"domains,":[13],"including":[14],"queueing":[15],"theory.":[16],"These":[17],"models":[18],"characterized":[20],"by":[21,57,176,277],"their":[22,299,302],"ability":[23],"to":[24,32,130,183,262,326],"maintain":[25],"complexity":[26],"at":[27],"low":[29],"level":[30],"due":[31],"property":[34],"known":[35],"as":[36,133,157,204,310],"the":[37,42,61,68,75,85,92,104,108,114,117,125,136,139,142,152,163,171,180,189,205,219,229,234,240,243,247,251,263,266,270,324],"property,":[39],"which":[40,59,307],"enables":[41,301],"connection":[43],"between":[44,318],"states":[45,115],"and":[46,63,161,207,250,321,335],"transition":[47,50,76,86,94,155,165],"probabilities.":[48],"The":[49,72,255,284,296,316],"matrices":[51],"of":[52,78,107,116,124,135,138,179,191,221,231,233,242,246,268,273,293,298],"represented":[56],"graphs,":[58],"show":[60],"properties":[62],"characteristics":[64],"that":[65,110,149,288],"help":[66],"analyze":[67],"underlying":[69],"processes.":[70],"Method":[71],"graph":[73,109,172],"representing":[74,91],"matrix":[77,156],"chain":[81,119,154,249,280],"is":[82,99,128,173,210,214],"formed":[83],"from":[84],"state":[87,164],"diagram,":[88],"with":[89,151],"weights":[90,137],"mean":[93],"rates.":[95],"A":[96,121,146],"probability":[97],"space":[98],"thus":[100],"created,":[101],"containing":[102],"all":[103],"spanning":[105,201],"trees":[106],"end":[111],"up":[112],"(anti-arborescences).":[120],"successive":[122],"examination":[123],"graph\u2019s":[126],"vertices":[127,181],"initiated":[129],"form":[131],"monomials":[132],"products":[134],"edges":[140],"forming":[141,200],"symbolic":[143],"solution.":[144],"Results":[145],"general":[147],"algorithm":[148,198,213],"commences":[150],"an":[158,192],"input":[159],"element":[160],"forms":[162],"diagram.":[166],"Subsequently,":[167],"each":[168,225],"vertex":[169],"within":[170],"examined,":[174],"followed":[175],"rearrangement":[178],"according":[182],"depth-first":[185],"search":[186],"strategy.":[187],"In":[188],"context":[190],"inverted":[193],"graph,":[194],"implementing":[195],"suitable":[197],"trees,":[202],"such":[203],"Gabow":[206],"Myers":[208],"algorithm,":[209],"imperative.":[211],"This":[212],"applied":[215],"sequentially,":[216],"resulting":[217],"formation":[220],"monomials,":[222],"polynomials":[223,232,238],"vertex,":[226],"and,":[227],"ultimately,":[228],"set":[230],"graph.":[235],"Utilizing":[236],"these":[237,289],"facilitates":[239],"calculation":[241],"stationary":[244],"probabilities":[245],"performance":[252,271],"metrics.":[253],"Discussion":[254],"proposed":[256],"method":[257],"provides":[258],"positive":[260],"response":[261],"inquiry":[264],"regarding":[265],"feasibility":[267],"expressing":[269],"metrics":[272],"system":[275],"modeled":[276],"through":[281],"closed-form":[282],"equations.":[283],"study":[285],"further":[286],"posits":[287],"specific":[290],"equations":[291],"considerable":[294],"magnitude.":[295],"intricacy":[297],"formulation":[300],"implementation":[303],"smaller":[305],"systems,":[306],"can":[308],"serve":[309],"building":[311],"blocks":[312],"other":[314],"methodologies.":[315],"correlation":[317],"graphs":[322],"has":[323],"potential":[325],"catalyze":[327],"novel":[328],"research":[329],"directions":[330],"both":[332],"discrete":[333],"mathematics":[334],"artificial":[336],"intelligence.":[337]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2025-10-10T00:00:00"}