Monotonicity Properties for Multi-Class Queueing Systems

<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="No"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>Monotonicity Properties for Multi-Class Queueing Systems | Discrete Event Dynamic Systems </title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="Monotonicity Properties for Multi-Class Queueing Systems"/> <meta name="twitter:description" content="Discrete Event Dynamic Systems - We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of..."/> <meta name="twitter:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs10626-009-0069-4/MediaObjects/10626_2009_69_Fig1_HTML.gif"/> <meta name="journal_id" content="10626"/> <meta name="dc.title" content="Monotonicity Properties for Multi-Class Queueing Systems"/> <meta name="dc.source" content="Discrete Event Dynamic Systems 2009 20:4"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2009-05-12"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2009 Springer Science+Business Media, LLC"/> <meta name="dc.rights" content="2009 Springer Science+Business Media, LLC"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted α-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter α and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well."/> <meta name="prism.issn" content="1573-7594"/> <meta name="prism.publicationName" content="Discrete Event Dynamic Systems"/> <meta name="prism.publicationDate" content="2009-05-12"/> <meta name="prism.volume" content="20"/> <meta name="prism.number" content="4"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="473"/> <meta name="prism.endingPage" content="509"/> <meta name="prism.copyright" content="2009 Springer Science+Business Media, LLC"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1007/s10626-009-0069-4"/> <meta name="prism.doi" content="doi:10.1007/s10626-009-0069-4"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1007/s10626-009-0069-4.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1007/s10626-009-0069-4"/> <meta name="citation_journal_title" content="Discrete Event Dynamic Systems"/> <meta name="citation_journal_abbrev" content="Discrete Event Dyn Syst"/> <meta name="citation_publisher" content="Springer US"/> <meta name="citation_issn" content="1573-7594"/> <meta name="citation_title" content="Monotonicity Properties for Multi-Class Queueing Systems"/> <meta name="citation_volume" content="20"/> <meta name="citation_issue" content="4"/> <meta name="citation_publication_date" content="2010/12"/> <meta name="citation_online_date" content="2009/05/12"/> <meta name="citation_firstpage" content="473"/> <meta name="citation_lastpage" content="509"/> <meta name="citation_article_type" content="Article"/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1007/s10626-009-0069-4"/> <meta name="DOI" content="10.1007/s10626-009-0069-4"/> <meta name="size" content="417813"/> <meta name="citation_doi" content="10.1007/s10626-009-0069-4"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1007/s10626-009-0069-4&api_key="/> <meta name="description" content="We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. 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The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted α-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter α and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well."/> <meta property="og:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs10626-009-0069-4/MediaObjects/10626_2009_69_Fig1_HTML.gif"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/oscar-static/img/favicons/darwin/favicon-c6d59aafac.ico> <meta name="theme-color" content="#e6e6e6">   <link 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The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted α-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter α and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well.","datePublished":"2009-05-12T00:00:00Z","dateModified":"2009-05-12T00:00:00Z","pageStart":"473","pageEnd":"509","sameAs":"https://doi.org/10.1007/s10626-009-0069-4","keywords":["Multi-class queueing systems","Sample-path comparisons","Monotonicity","Mean number of users","Bandwidth-sharing networks","Weighted α-fair policies","cμ-rule","DPS","GPS","Systems Theory","Control","Operations Research/Decision Theory","Convex and Discrete Geometry","Electrical Engineering","Manufacturing","Machines","Tools","Processes"],"image":["https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10626-009-0069-4/MediaObjects/10626_2009_69_Fig1_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10626-009-0069-4/MediaObjects/10626_2009_69_Fig2_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10626-009-0069-4/MediaObjects/10626_2009_69_Fig3_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10626-009-0069-4/MediaObjects/10626_2009_69_Fig4_HTML.gif","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs10626-009-0069-4/MediaObjects/10626_2009_69_Fig5_HTML.gif"],"isPartOf":{"name":"Discrete Event Dynamic Systems","issn":["1573-7594","0924-6703"],"volumeNumber":"20","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer US","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Ina Maria Verloop","affiliation":[{"name":"CWI","address":{"name":"CWI, Amsterdam, The Netherlands","@type":"PostalAddress"},"@type":"Organization"}],"email":"maaike@cwi.nl","@type":"Person"},{"name":"Urtzi Ayesta","affiliation":[{"name":"CNRS","address":{"name":"LAAS, CNRS, Toulouse Cedex, France","@type":"PostalAddress"},"@type":"Organization"},{"name":"BCAM – Basque Center for Applied Mathematics","address":{"name":"BCAM – Basque Center for Applied Mathematics, Zamudio, Spain","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Sem Borst","affiliation":[{"name":"CWI","address":{"name":"CWI, Amsterdam, The Netherlands","@type":"PostalAddress"},"@type":"Organization"},{"name":"Alcatel-Lucent","address":{"name":"Bell Laboratories, Alcatel-Lucent, Murray Hill, USA","@type":"PostalAddress"},"@type":"Organization"},{"name":"Eindhoven University of Technology","address":{"name":"Department of Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"}],"isAccessibleForFree":false,"hasPart":{"isAccessibleForFree":false,"cssSelector":".main-content","@type":"WebPageElement"},"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" >  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>   <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-visually-hidden" aria-hidden="true" data-test="darwin-icons"> <?xml version="1.0" 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href="https://link.springer.com/journal/10626/aims-and-scope" class="app-article-masthead__submission-link" data-track="click_aims_and_scope" data-track-action="aims and scope" data-track-context="article page" data-track-label="link"> Aims and scope <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> <a href="https://submission.springernature.com/new-submission/10626/3" class="app-article-masthead__submission-link" data-track="click_submit_manuscript" data-track-context="article masthead on springerlink article page" data-track-action="submit manuscript" data-track-label="link"> Submit manuscript <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> </div> </div> </div> </section> <div class="c-article-main u-container u-mt-24 u-mb-32 l-with-sidebar" id="main-content" data-component="article-container"> <main class="u-serif js-main-column" data-track-component="article body"> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Ina_Maria-Verloop-Aff1" data-author-popup="auth-Ina_Maria-Verloop-Aff1" data-author-search="Verloop, Ina Maria" data-corresp-id="c1">Ina Maria Verloop<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Urtzi-Ayesta-Aff2-Aff3" data-author-popup="auth-Urtzi-Ayesta-Aff2-Aff3" data-author-search="Ayesta, Urtzi">Urtzi Ayesta</a><sup class="u-js-hide"><a href="#Aff2">2</a>,<a href="#Aff3">3</a></sup> & </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Sem-Borst-Aff1-Aff4-Aff5" data-author-popup="auth-Sem-Borst-Aff1-Aff4-Aff5" data-author-search="Borst, Sem">Sem Borst</a><sup class="u-js-hide"><a href="#Aff1">1</a>,<a href="#Aff4">4</a>,<a href="#Aff5">5</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use 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class="c-article-section__content" id="Abs1-content"><p>We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted <i>α</i>-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter <i>α</i> and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well.</p></div></div></section> <div class="c-notes"> <p class="c-notes__text c-status-message--info"> <svg width="24" height="24" focusable="false" role="img" aria-hidden="true" class="c-status-message__icon"> <use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-info-filled-medium"></use> </svg> This is a preview of subscription content, <a id="test-login-banner-link" href="//wayf.springernature.com?redirect_uri=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs10626-009-0069-4%3Ferror%3Dcookies_not_supported%26code%3D16885cf7-d107-41fb-b690-7c234dd743a3" data-track="click" data-track-action="login" data-track-label="link" class="c-preview-message__link">log in via an institution</a> <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon c-external-link__icon"> <use 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data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/s10479-015-2060-9">Analysis of discrete-time queues with general service demands and finite-support service capacities </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">17 November 2015</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1743014950, embedded_user: 'null' } }); </script> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" 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In: Proceedings of ACM SIGMETRICS</p></li></ul><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s10626-009-0069-4?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p>The authors are grateful to Matthieu Jonckheere (Eindhoven University of Technology, The Netherlands) for helpful discussions on stochastic comparisons.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">CWI, P.O. Box 94079, 1090 GB, Amsterdam, The Netherlands</p><p class="c-article-author-affiliation__authors-list">Ina Maria Verloop & Sem Borst</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">LAAS, CNRS, 7 Avenue Colonel Roche, 31077, Toulouse Cedex, France</p><p class="c-article-author-affiliation__authors-list">Urtzi Ayesta</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">BCAM – Basque Center for Applied Mathematics, Bizkaia Technology Park, 48170, Zamudio, Spain</p><p class="c-article-author-affiliation__authors-list">Urtzi Ayesta</p></li><li id="Aff4"><p class="c-article-author-affiliation__address">Bell Laboratories, Alcatel-Lucent, P.O. Box 636, Murray Hill, NJ, 07974, USA</p><p class="c-article-author-affiliation__authors-list">Sem Borst</p></li><li id="Aff5"><p class="c-article-author-affiliation__address">Department of Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands</p><p class="c-article-author-affiliation__authors-list">Sem Borst</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Ina_Maria-Verloop-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Ina Maria Verloop</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?sortBy=newestFirst&dc.creator=Ina%20Maria%20Verloop" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text"><span class="c-article-authors-search__links-text">You can also search for this author in</span><span class="c-article-identifiers"><a class="c-article-identifiers__item" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Ina%20Maria%20Verloop" data-track="click" data-track-action="author link - 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pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Urtzi%20Ayesta%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-Sem-Borst-Aff1-Aff4-Aff5"><span class="c-article-authors-search__title u-h3 js-search-name">Sem Borst</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?sortBy=newestFirst&dc.creator=Sem%20Borst" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text"><span class="c-article-authors-search__links-text">You can also search for this author in</span><span class="c-article-identifiers"><a class="c-article-identifiers__item" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Sem%20Borst" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Sem%20Borst%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:maaike@cwi.nl">Ina Maria Verloop</a>.</p></div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><p>A shorter version with preliminary results appeared in the proceedings of ValueTools (Verloop et al. <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2008" title="Verloop IM, Ayesta U, Borst SC (2008) Comparison of bandwidth-sharing policies in a linear network. In: Proceedings of ValueTools, Athens" href="/article/10.1007/s10626-009-0069-4#ref-CR38" id="ref-link-section-d21718553e486">2008</a>).</p></div></div></section><section aria-labelledby="appendices"><div class="c-article-section" id="appendices-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="appendices">Appendices</h2><div class="c-article-section__content" id="appendices-content"><h3 class="c-article__sub-heading" id="App1">Appendix A: Proof of Lemma 5.1</h3><p>For a given state <span class="mathjax-tex">$\vec{n}$</span>, the <i>α</i>-fair allocation is the vector (<i>s</i> <sub>0</sub>,<i>s</i> <sub>1</sub>,..., <i>s</i> <sub> <i>L</i> </sub>) that solves the optimization problem (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ7">7</a>). If <i>n</i> <sub> <i>i</i> </sub> > 0, then <i>s</i> <sub> <i>i</i> </sub> = <i>C</i> <sub> <i>i</i> </sub> − <i>s</i> <sub>0</sub>. The objective function in Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ7">7</a> expressed in terms of the value <i>s</i> <sub>0</sub>, is concave in <i>s</i> <sub>0</sub>. Taking the derivative of Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ7">7</a> with respect to <i>s</i> <sub>0</sub> and setting it equal to zero, we obtain that <span class="mathjax-tex">$s_0^{\pi(\alpha,w)}(\vec{n})$</span> satisfies </p><div id="Equx" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ w_0\cdot n_0^\alpha \cdot (s_0^{\pi(\alpha, w)}(\vec{n}))^{-\alpha} = \sum\limits_{i=1}^L w_i\cdot n_i^\alpha \cdot (C_i-s_0^{\pi(\alpha, w)}(\vec{n}))^{-\alpha}, $$</span></div></div><p> or equivalently </p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ 1=\sum\limits_{i=1}^L\frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0} \frac{s_0^{\pi(\alpha, w)}(\vec{n})}{C_i-s_0^{\pi(\alpha, w)}(\vec{n})}\Bigr)^\alpha. $$</span></div><div class="c-article-equation__number"> (28) </div></div><p>The function <span class="mathjax-tex">$\sum_{i=1}^L \frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0}{C_i-s_0}\Bigr)^\alpha$</span> is non-decreasing in <i>s</i> <sub>0</sub>. Hence, when either <i>n</i> <sub> <i>i</i> </sub> or <span class="mathjax-tex">$\frac{w_i}{w_0}$</span> increases, by equality (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ28">28</a>) the corresponding value of <i>s</i> <sub>0</sub> must decrease. Statements (i) and (iii) follow now immediately.</p><p>Statement (ii) follows similarly by noting that </p><div id="Equy" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{array}{lll} \Bigl(\sum\limits_{i=1}^L \frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\gamma,w)}(\vec{n})}{C_i-s^{\pi(\gamma,w)}_0(\vec{n})}\Bigr)^\gamma \Bigl)^{\frac{1}{\gamma}} = 1&=& \Bigl(\sum\limits_{i=1}^L \frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\beta,w)}(\vec{n})}{C_i-s^{\pi(\beta,w)}_0(\vec{n})}\Bigr)^\beta \Bigl)^{\frac{1}{\beta}} \\[4pt] &=&\Bigr(\sum\limits_{i=1}^L\frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\beta,w)}(\vec{n})}{C_i-s^{\pi(\beta,w)}_0(\vec{n})}\Bigr)^\beta \Bigl)^{\frac{r}{r\beta}}\\ & \geq& \Bigr(\sum\limits_{i=1}^L\frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\beta,w)}(\vec{n})}{C_i-s^{\pi(\beta,w)}_0(\vec{n})}\Bigr)^{r\beta} \Bigl)^{\frac{1}{r\beta}} \\&=& \Bigr(\sum\limits_{i=1}^L\frac{w_i}{w_0} \Bigl(\frac{n_i}{n_0}\frac{s_0^{\pi(\beta,w)}(\vec{n})}{C_i-s_0^{\pi(\beta,w)}(\vec{n})}\Bigr)^\gamma \Bigl)^{\frac{1}{\gamma}}, \end{array}$$</span></div></div><p>with <i>rβ</i> = <i>γ</i> and <i>r</i> > 1. Hence <span class="mathjax-tex">$s^{\pi(\beta,w)}_0(\vec{n})\leq s^{\pi(\gamma,w)}_0(\vec{n})$</span>.□</p><h3 class="c-article__sub-heading" id="App2">Appendix B: Proof of Proposition 5.6</h3><p>By the conjecture of Kang et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2009" title="Kang WN, Kelly FP, Lee NH, Williams RJ (2009) State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann Appl Probab (in press)" href="/article/10.1007/s10626-009-0069-4#ref-CR14" id="ref-link-section-d21718553e10700">2009</a>), the scaled workload in a node is independent of <i>α</i>. In addition, it is stated that the diffusion scaled number of users, <span class="mathjax-tex">$\vec{\hat N}^{k,\pi(\alpha)}(t)$</span>, converges in distribution as <i>k</i>→ ∞ to <span class="mathjax-tex">$\vec{\hat N}^{\pi(\alpha)}(t)=\Delta(\vec{\hat{V}}^{\pi(\alpha)}(t))$</span>, where the lifting mapping <span class="mathjax-tex">$\Delta: \mathbf{R}_+^2 \to \mathbf{R}_+^3$</span> is as defined in Kang et al. (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2009" title="Kang WN, Kelly FP, Lee NH, Williams RJ (2009) State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann Appl Probab (in press)" href="/article/10.1007/s10626-009-0069-4#ref-CR14" id="ref-link-section-d21718553e10743">2009</a>) and Kelly and Williams (<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2004" title="Kelly FP, Williams RJ (2004) Fluid model for a network operating under a fair bandwidth-sharing policy. Ann Appl Probab 14:1055–1083" href="/article/10.1007/s10626-009-0069-4#ref-CR17" id="ref-link-section-d21718553e10746">2004</a>). This is equivalent to saying that there are <i>q</i> <sub>1</sub>(<i>α</i>), <i>q</i> <sub>2</sub>(<i>α</i>) ≥ 0 such that </p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \label{eq:dif_n} \hat N_i^{\pi(\alpha)} = \rho_iq_i(\alpha)^{\frac{1}{\alpha}}, \ \ i=1,2 \ \ \mbox{ and } \ \ \hat N_0^{\pi(\alpha)}=\rho_0(q_1(\alpha)+q_2(\alpha))^{\frac{1}{\alpha}}. $$</span></div><div class="c-article-equation__number"> (29) </div></div><p>Using this representation for the number of users, we can describe the effect the parameter <i>α</i> has on the holding cost.</p><p>We compare the holding cost under two <i>α</i>-fair policies with parameters <i>α</i> <sub>1</sub> and <i>α</i> <sub>2</sub> for a given workload in both nodes. So we have at each time that the workload in a node is <span class="mathjax-tex">$\hat V_i^{\pi(\alpha)}=\hat v_i,$</span> independent of <i>α</i>, <i>i</i> = 1,2 (from now on we will drop the dependence on <i>t</i>). Using the representation as in Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ9">9</a> and the fact that <i>ρ</i> <sub>0</sub> + <i>ρ</i> <sub> <i>i</i> </sub> = <i>C</i> <sub> <i>i</i> </sub>, this gives </p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ (C_i-\rho_0)q_i(\alpha)^{1/\alpha}+\rho_0\frac{\mu_i}{\mu_0}(q_1(\alpha)+q_2(\alpha))^{1/\alpha} =\hat v_i. $$</span></div><div class="c-article-equation__number"> (30) </div></div><p>Together with Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ29">29</a>, the holding cost for an <i>α</i>-fair policy can be written as </p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{array}{lll} \sum\limits_{i=0}^2 c_i \hat N_i^{\pi(\alpha)} &=& c_0\rho_0(q_1(\alpha)+q_2(\alpha))^{\frac{1}{\alpha}} +c_1(C_1-\rho_0) q_1(\alpha)^{\frac{1}{\alpha}} +c_2(C_2-\rho_0)q_2(\alpha)^{\frac{1}{\alpha}} \\ &=& c_1 \Bigl( (C_1-\rho_0)q_1(\alpha)^{1/\alpha}+\rho_0\frac{\mu_1}{\mu_0}(q_1(\alpha)+q_2(\alpha))^{1/\alpha}\Bigr) \\ && +\,c_2\Bigl((C_2-\rho_0)q_2(\alpha)^{1/\alpha}+\rho_0\frac{\mu_2}{\mu_0}(q_1(\alpha)+q_2(\alpha))^{1/\alpha}\Bigr) \\ && +\, \frac{c_0\mu_0-c_1\mu_1-c_2\mu_2}{\mu_0}\rho_0(q_1(\alpha)+q_2(\alpha))^{\frac{1}{\alpha}} \\ &\stackrel{d}{=}& c_1 \hat v_1+ c_2\hat v_2 +\frac{c_0\mu_0-c_1\mu_1-c_2\mu_2}{\mu_0}\rho_0(1+f(\alpha)^\alpha)^{\frac{1}{\alpha}}q_2(\alpha)^{\frac{1}{\alpha}}, \end{array}$$</span></div><div class="c-article-equation__number"> (31) </div></div><p> where <span class="mathjax-tex">$f(\alpha):=\Bigl(\frac{q_1(\alpha)}{q_2(\alpha)}\Bigr)^\frac{1}{\alpha}$</span>.</p><p>For <i>i</i> = 2, Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ30">30</a> gives </p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{array}{lll} &&{\kern-6pt} q_2(\alpha_1)^{1/\alpha_1}\Bigl(C_2-\rho_0+\rho_0\frac{\mu_2}{\mu_0}(1+f(\alpha_1)^{\alpha_1})^{1/\alpha_1}\Bigr)\\ &&{\kern4pt} =q_2(\alpha_2)^{1/\alpha_2}\Bigl(C_2-\rho_0+\rho_0\frac{\mu_2}{\mu_0}(1+f(\alpha_2)^{\alpha_2})^{1/\alpha_2}\Bigr). \label{i2} \end{array}$$</span></div><div class="c-article-equation__number"> (32) </div></div><p> From Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ32">32</a> we conclude that </p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ (1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}q_2(\alpha_1)^{\frac{1}{\alpha_1}} < \ (=) \ (1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}q_2(\alpha_2)^{\frac{1}{\alpha_2}} $$</span></div><div class="c-article-equation__number"> (33) </div></div><p>if and only if </p><div id="Equz" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\frac{(1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}}{(C_2-\rho_0+\rho_0\frac{\mu_2}{\mu_0}(1+f(\alpha_1)^{\alpha_1})^{1/\alpha_1}} < \ (=) \ \frac{(1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}}{(C_2-\rho_0+\rho_0\frac{\mu_2}{\mu_0}(1+f(\alpha_2)^{\alpha_2})^{1/\alpha_2}}, $$</span></div></div><p> if and only if </p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ (1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}< \ (=) \ (1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}. $$</span></div><div class="c-article-equation__number"> (34) </div></div> <p>Let <i>b</i> be such that <span class="mathjax-tex">$\hat v_1=b\hat v_2$</span>. Assume without loss of generality <span class="mathjax-tex">$\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0} \leq b\leq 1$</span>. (For states with <i>b</i> > 1 the analysis is the same, with only the roles of nodes 1 and 2 interchanged.) Note that when <span class="mathjax-tex">$b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}$</span> we are on the edge of the cone as described in Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ9">10</a>. In Lemma 1 (see below) we prove that <span class="mathjax-tex">$(1+f(\alpha)^{\alpha})^{\frac{1}{\alpha}}$</span> is indeed strictly decreasing in <i>α</i> when <span class="mathjax-tex">$\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}< b \leq 1$</span> and is constant when <span class="mathjax-tex">$b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}$</span>, the edge of the cone. Assuming the probability mass is not all concentrated on the edge of the cone, we conclude from Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ31">31</a> and the equivalence between Eqs. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ33">33</a> and <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ34">34</a>, that the mean holding cost is strictly decreasing (strictly increasing) in <i>α</i> when <i>c</i> <sub>1</sub> <i>μ</i> <sub>1</sub> + <i>c</i> <sub>2</sub> <i>μ</i> <sub>2</sub> < <i>c</i> <sub>0</sub> <i>μ</i> <sub>0</sub> (<i>c</i> <sub>1</sub> <i>μ</i> <sub>1</sub> + <i>c</i> <sub>2</sub> <i>μ</i> <sub>2</sub> > <i>c</i> <sub>0</sub> <i>μ</i> <sub>0</sub>). □</p><p>The following lemma is used in the proof of Proposition 5.6.</p> <h3 class="c-article__sub-heading"> <b>Lemma 1</b> </h3> <p><i>The function </i>(1 + <i>f</i>(<i>α</i>)<sup><i>α</i></sup>)<sup>1/<i>α</i></sup><i> with </i><span class="mathjax-tex">$f(\alpha)=\Bigl(\frac{q_1(\alpha)}{q_2(\alpha)}\Bigr)^{\frac{1}{\alpha}}$</span><i>, is strictly decreasing in α</i><i> when </i><span class="mathjax-tex">$\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0} < b\leq 1$</span><i> and is constant when </i><span class="mathjax-tex">$b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}.$</span><i> Here b</i><i> satisfies </i><span class="mathjax-tex">$\hat v_1=b\hat v_2$</span><i>.</i></p> <h3 class="c-article__sub-heading">Proof</h3> <p>From <span class="mathjax-tex">$\hat v_1=b\hat v_2$</span> and Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ30">30</a> we obtain the relation </p><div id="Equaa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ (C_1-\rho_0)q_1(\alpha_i)^{\frac{1}{\alpha_i}}+(1-b)\rho_0\frac{\mu_1}{\mu_0}(q_1(\alpha_i)+q_2(\alpha_i))^{\frac{1}{\alpha_i}}=b(C_2-\rho_0)\frac{\mu_1}{\mu_2}q_2(\alpha_i)^{\frac{1}{\alpha_i}}, $$</span></div></div><p>hence when we divide both sides by <span class="mathjax-tex">$q_2(\alpha_i)^{\frac{1}{\alpha_i}}$</span>, we obtain </p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ (C_1-\rho_0)f(\alpha_i)+(1-b)\rho_0\frac{\mu_1}{\mu_0}(1+f(\alpha_i)^{\alpha_i})^{\frac{1}{\alpha_i}}=b(C_2-\rho_0)\frac{\mu_1}{\mu_2}. $$</span></div><div class="c-article-equation__number"> (35) </div></div> <p>By Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ35">35</a> we have that <i>f</i>(<i>α</i>) = 0 if and only if <span class="mathjax-tex">$b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}$</span>.</p> <p>Assume <span class="mathjax-tex">$b=\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0}$</span>. Then <i>f</i>(<i>α</i>) = 0 and hence the function <span class="mathjax-tex">$(1+f(\alpha)^{\alpha})^{\frac{1}{\alpha}}$</span> is constant.</p> <p>Now assume <span class="mathjax-tex">$\frac{\rho_0/\mu_0}{(C_2-\rho_0)/\mu_2+\rho_0/\mu_0} < b\leq 1$</span>. So <i>f</i>(<i>α</i>) > 0 for all <i>α</i>. Take <i>α</i> <sub>1</sub> < <i>α</i> <sub>2</sub> and let <i>r</i> > 1 be such that <i>α</i> <sub>2</sub> = <i>rα</i> <sub>1</sub>. Then </p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$(1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}} = (1^{r\alpha_1}+f(r\alpha_1)^{r\alpha_1})^{\frac{1}{r\alpha_1}} <(1^{\alpha_1}+f(r\alpha_1)^{\alpha_1})^{\frac{r}{r\alpha_1}} =(1+f(r\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}, $$</span></div><div class="c-article-equation__number"> (36) </div></div><p>since 1 + <i>f</i>(<i>rα</i> <sub>1</sub>) > 1. Suppose <i>f</i>(<i>α</i> <sub>2</sub>) = <i>f</i>(<i>rα</i> <sub>1</sub>) ≤ <i>f</i>(<i>α</i> <sub>1</sub>). From Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ36">36</a>, we then obtain <span class="mathjax-tex">$(1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}< (1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}$</span>. However, from Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ35">35</a> we know that when <i>f</i>(<i>α</i> <sub>2</sub>) ≤ <i>f</i>(<i>α</i> <sub>1</sub>), then <span class="mathjax-tex">$(1+f(\alpha_2)^{\alpha_2})^{\frac{1}{\alpha_2}}\geq (1+f(\alpha_1)^{\alpha_1})^{\frac{1}{\alpha_1}}$</span>, hence we have a contradiction. So we conclude that <i>f</i>(<i>α</i> <sub>2</sub>) > <i>f</i>(<i>α</i> <sub>1</sub>), and hence <i>f</i>(<i>α</i>) is strictly increasing in <i>α</i> and from Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10626-009-0069-4#Equ35">35</a> it then follows that <span class="mathjax-tex">$(1+f(\alpha)^\alpha)^\frac{1}{\alpha}$</span> is strictly decreasing in <i>α</i>.□</p> </div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"><p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=Monotonicity%20Properties%20for%20Multi-Class%20Queueing%20Systems&author=Ina%20Maria%20Verloop%20et%20al&contentID=10.1007%2Fs10626-009-0069-4&copyright=Springer%20Science%2BBusiness%20Media%2C%20LLC&publication=0924-6703&publicationDate=2009-05-12&publisherName=SpringerNature&orderBeanReset=true">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="c-bibliographic-information__column"><h3 class="c-article__sub-heading" id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Verloop, I.M., Ayesta, U. & Borst, S. 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Monotonicity Properties for Multi-Class Queueing Systems | Discrete Event Dynamic Systems