An iterative approximation scheme for repetitive Markov processes
Repetitive Markov processes form a class of processes where the generator matrix has a particular repeating form. Many queueing models fall in this category such as M/M/1 queues, quasi-birth-and-death processes, and processes with M/G/1 or GI/M/1 generator matrices. In this paper, a new iterative sc...
Gespeichert in:
| Veröffentlicht in: | Journal of applied probability 1999-09, Vol.36 (3), p.654-667 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 667 |
|---|---|
| container_issue | 3 |
| container_start_page | 654 |
| container_title | Journal of applied probability |
| container_volume | 36 |
| creator | Tufecki, Tolga Gullu, Refik |
| description | Repetitive Markov processes form a class of processes where the generator matrix has a particular repeating form. Many queueing models fall in this category such as M/M/1 queues, quasi-birth-and-death processes, and processes with M/G/1 or GI/M/1 generator matrices. In this paper, a new iterative scheme is proposed for computing the stationary probabilities of such processes. An infinite state process is approximated by a finite state process by lumping an infinite number of states into a super-state. What we call the feedback rate, the conditional expected rate of flow from the super-state to the remaining states, given the process is in the super-state, is approximated simultaneously with the steady state probabilities. The method is theoretically developed and numerically tested for quasi-birth-and-death processes. It turns out that the new concept of the feedback rate can be effectively used in computing the stationary probabilities. |
| doi_str_mv | 10.1239/jap/1032374624 |
| format | Article |
| fullrecord | <record><control><sourceid>jstor_proje</sourceid><recordid>TN_cdi_projecteuclid_primary_oai_CULeuclid_euclid_jap_1032374624</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><cupid>10_1239_jap_1032374624</cupid><jstor_id>3215431</jstor_id><sourcerecordid>3215431</sourcerecordid><originalsourceid>FETCH-LOGICAL-c369t-fa53fb3aafce562c212267a78ee96050b944bb61af8d067c703524bd5cec9e3f3</originalsourceid><addsrcrecordid>eNp1kDtPwzAUhS0EEqWwMjFEiDWtX7HxRlXxkopY6Bw5zjU4tHWw0wr-PYZG7YBYfGXfz-ccHYTOCR4RytS40e2YYEaZ5ILyAzQgXBa5wJIeogHGlOQqncfoJMYGY8ILJQdoMlllroOgO7eBTLdt8J9umW5-lUXzBkvIrA9ZgBY698s86fDuN1kCDcQI8RQdWb2IcNbPIZrf3b5MH_LZ8_3jdDLLDROqy60umK2Y1tZAIaihhFIhtbwGUAIXuFKcV5Ug2l7XWEgjMSsor-rCgFHALBuim61ucm7AdLA2C1eXbUhxw1fptSun81n_2o9USbmvJElc7iQ-1hC7svHrsEqpS8q4woxLkaDRFjLBxxjA7iwILn-K_qt61avqaPTCBr0yLu5_US4VZwm72GJN7HzYrRklBWckrXFvq5dVcPUr7MP9Y_wN0p-XhQ</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>234903476</pqid></control><display><type>article</type><title>An iterative approximation scheme for repetitive Markov processes</title><source>JSTOR Mathematics & Statistics</source><source>JSTOR Archive Collection A-Z Listing</source><creator>Tufecki, Tolga ; Gullu, Refik</creator><creatorcontrib>Tufecki, Tolga ; Gullu, Refik</creatorcontrib><description>Repetitive Markov processes form a class of processes where the generator matrix has a particular repeating form. Many queueing models fall in this category such as M/M/1 queues, quasi-birth-and-death processes, and processes with M/G/1 or GI/M/1 generator matrices. In this paper, a new iterative scheme is proposed for computing the stationary probabilities of such processes. An infinite state process is approximated by a finite state process by lumping an infinite number of states into a super-state. What we call the feedback rate, the conditional expected rate of flow from the super-state to the remaining states, given the process is in the super-state, is approximated simultaneously with the steady state probabilities. The method is theoretically developed and numerically tested for quasi-birth-and-death processes. It turns out that the new concept of the feedback rate can be effectively used in computing the stationary probabilities.</description><identifier>ISSN: 0021-9002</identifier><identifier>EISSN: 1475-6072</identifier><identifier>DOI: 10.1239/jap/1032374624</identifier><identifier>CODEN: JPRBAM</identifier><language>eng</language><publisher>Cambridge, UK: Cambridge University Press</publisher><subject>60J05 ; 60J10 ; 60K25 ; approximate solutions ; Approximation ; Conditional probabilities ; Eigenvalues ; Eigenvectors ; Equations ; Exact sciences and technology ; Markov analysis ; Markov chains ; Markov processes ; Mathematical vectors ; Mathematics ; Matrices ; Nontrivial solutions ; numerical algorithms ; Probability ; Probability and statistics ; Probability theory and stochastic processes ; Quasi-birth-and-death processes ; Queuing theory ; Research Papers ; Sciences and techniques of general use ; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications) ; Studies</subject><ispartof>Journal of applied probability, 1999-09, Vol.36 (3), p.654-667</ispartof><rights>Copyright © Applied Probability Trust 1999</rights><rights>Copyright 1999 Applied Probability Trust</rights><rights>2000 INIST-CNRS</rights><rights>Copyright Applied Probability Trust Sep 1999</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c369t-fa53fb3aafce562c212267a78ee96050b944bb61af8d067c703524bd5cec9e3f3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/3215431$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/3215431$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,780,784,803,832,885,27924,27925,58017,58021,58250,58254</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=1247943$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Tufecki, Tolga</creatorcontrib><creatorcontrib>Gullu, Refik</creatorcontrib><title>An iterative approximation scheme for repetitive Markov processes</title><title>Journal of applied probability</title><addtitle>Journal of Applied Probability</addtitle><description>Repetitive Markov processes form a class of processes where the generator matrix has a particular repeating form. Many queueing models fall in this category such as M/M/1 queues, quasi-birth-and-death processes, and processes with M/G/1 or GI/M/1 generator matrices. In this paper, a new iterative scheme is proposed for computing the stationary probabilities of such processes. An infinite state process is approximated by a finite state process by lumping an infinite number of states into a super-state. What we call the feedback rate, the conditional expected rate of flow from the super-state to the remaining states, given the process is in the super-state, is approximated simultaneously with the steady state probabilities. The method is theoretically developed and numerically tested for quasi-birth-and-death processes. It turns out that the new concept of the feedback rate can be effectively used in computing the stationary probabilities.</description><subject>60J05</subject><subject>60J10</subject><subject>60K25</subject><subject>approximate solutions</subject><subject>Approximation</subject><subject>Conditional probabilities</subject><subject>Eigenvalues</subject><subject>Eigenvectors</subject><subject>Equations</subject><subject>Exact sciences and technology</subject><subject>Markov analysis</subject><subject>Markov chains</subject><subject>Markov processes</subject><subject>Mathematical vectors</subject><subject>Mathematics</subject><subject>Matrices</subject><subject>Nontrivial solutions</subject><subject>numerical algorithms</subject><subject>Probability</subject><subject>Probability and statistics</subject><subject>Probability theory and stochastic processes</subject><subject>Quasi-birth-and-death processes</subject><subject>Queuing theory</subject><subject>Research Papers</subject><subject>Sciences and techniques of general use</subject><subject>Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications)</subject><subject>Studies</subject><issn>0021-9002</issn><issn>1475-6072</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1999</creationdate><recordtype>article</recordtype><recordid>eNp1kDtPwzAUhS0EEqWwMjFEiDWtX7HxRlXxkopY6Bw5zjU4tHWw0wr-PYZG7YBYfGXfz-ccHYTOCR4RytS40e2YYEaZ5ILyAzQgXBa5wJIeogHGlOQqncfoJMYGY8ILJQdoMlllroOgO7eBTLdt8J9umW5-lUXzBkvIrA9ZgBY698s86fDuN1kCDcQI8RQdWb2IcNbPIZrf3b5MH_LZ8_3jdDLLDROqy60umK2Y1tZAIaihhFIhtbwGUAIXuFKcV5Ug2l7XWEgjMSsor-rCgFHALBuim61ucm7AdLA2C1eXbUhxw1fptSun81n_2o9USbmvJElc7iQ-1hC7svHrsEqpS8q4woxLkaDRFjLBxxjA7iwILn-K_qt61avqaPTCBr0yLu5_US4VZwm72GJN7HzYrRklBWckrXFvq5dVcPUr7MP9Y_wN0p-XhQ</recordid><startdate>19990901</startdate><enddate>19990901</enddate><creator>Tufecki, Tolga</creator><creator>Gullu, Refik</creator><general>Cambridge University Press</general><general>Applied Probability Trust</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>H8D</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>19990901</creationdate><title>An iterative approximation scheme for repetitive Markov processes</title><author>Tufecki, Tolga ; Gullu, Refik</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c369t-fa53fb3aafce562c212267a78ee96050b944bb61af8d067c703524bd5cec9e3f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1999</creationdate><topic>60J05</topic><topic>60J10</topic><topic>60K25</topic><topic>approximate solutions</topic><topic>Approximation</topic><topic>Conditional probabilities</topic><topic>Eigenvalues</topic><topic>Eigenvectors</topic><topic>Equations</topic><topic>Exact sciences and technology</topic><topic>Markov analysis</topic><topic>Markov chains</topic><topic>Markov processes</topic><topic>Mathematical vectors</topic><topic>Mathematics</topic><topic>Matrices</topic><topic>Nontrivial solutions</topic><topic>numerical algorithms</topic><topic>Probability</topic><topic>Probability and statistics</topic><topic>Probability theory and stochastic processes</topic><topic>Quasi-birth-and-death processes</topic><topic>Queuing theory</topic><topic>Research Papers</topic><topic>Sciences and techniques of general use</topic><topic>Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications)</topic><topic>Studies</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Tufecki, Tolga</creatorcontrib><creatorcontrib>Gullu, Refik</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal of applied probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Tufecki, Tolga</au><au>Gullu, Refik</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>An iterative approximation scheme for repetitive Markov processes</atitle><jtitle>Journal of applied probability</jtitle><addtitle>Journal of Applied Probability</addtitle><date>1999-09-01</date><risdate>1999</risdate><volume>36</volume><issue>3</issue><spage>654</spage><epage>667</epage><pages>654-667</pages><issn>0021-9002</issn><eissn>1475-6072</eissn><coden>JPRBAM</coden><abstract>Repetitive Markov processes form a class of processes where the generator matrix has a particular repeating form. Many queueing models fall in this category such as M/M/1 queues, quasi-birth-and-death processes, and processes with M/G/1 or GI/M/1 generator matrices. In this paper, a new iterative scheme is proposed for computing the stationary probabilities of such processes. An infinite state process is approximated by a finite state process by lumping an infinite number of states into a super-state. What we call the feedback rate, the conditional expected rate of flow from the super-state to the remaining states, given the process is in the super-state, is approximated simultaneously with the steady state probabilities. The method is theoretically developed and numerically tested for quasi-birth-and-death processes. It turns out that the new concept of the feedback rate can be effectively used in computing the stationary probabilities.</abstract><cop>Cambridge, UK</cop><pub>Cambridge University Press</pub><doi>10.1239/jap/1032374624</doi><tpages>14</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0021-9002 |
| ispartof | Journal of applied probability, 1999-09, Vol.36 (3), p.654-667 |
| issn | 0021-9002 1475-6072 |
| language | eng |
| recordid | cdi_projecteuclid_primary_oai_CULeuclid_euclid_jap_1032374624 |
| source | JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing |
| subjects | 60J05 60J10 60K25 approximate solutions Approximation Conditional probabilities Eigenvalues Eigenvectors Equations Exact sciences and technology Markov analysis Markov chains Markov processes Mathematical vectors Mathematics Matrices Nontrivial solutions numerical algorithms Probability Probability and statistics Probability theory and stochastic processes Quasi-birth-and-death processes Queuing theory Research Papers Sciences and techniques of general use Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications) Studies |
| title | An iterative approximation scheme for repetitive Markov processes |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-03T02%3A05%3A04IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_proje&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=An%20iterative%20approximation%20scheme%20for%20repetitive%20Markov%20processes&rft.jtitle=Journal%20of%20applied%20probability&rft.au=Tufecki,%20Tolga&rft.date=1999-09-01&rft.volume=36&rft.issue=3&rft.spage=654&rft.epage=667&rft.pages=654-667&rft.issn=0021-9002&rft.eissn=1475-6072&rft.coden=JPRBAM&rft_id=info:doi/10.1239/jap/1032374624&rft_dat=%3Cjstor_proje%3E3215431%3C/jstor_proje%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=234903476&rft_id=info:pmid/&rft_cupid=10_1239_jap_1032374624&rft_jstor_id=3215431&rfr_iscdi=true |