<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"       "http://www.w3.org/TR/REC-html40/loose.dtd">  <HTML> <HEAD><TITLE>RR-3398 : Thermodynamic Limit and Propagation of Chaos in Polling Networks </TITLE> <link rel="stylesheet" href="/css/pagestrict.css"> <link rel="stylesheet" href="/css/pagefinalestrict.css"> <META NAME="code" CONTENT="MEVAL">  <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <META NAME="description" CONTENT="Delcoigne, Franck / Fayolle, Guy"> <META NAME="dc.title" CONTENT="Thermodynamic Limit and Propagation of Chaos in Polling Networks"> <META NAME="dc.creator" CONTENT="Delcoigne, Franck; Fayolle, Guy"> <META NAME="dc.subject" CONTENT="PROCESSUS DE MARKOV; RSEAUX  SCRUTIN; LIMITE THERMODYNAMIQUE; CHAMP MOYEN; PROPAGATION DU CHAOS; MARKOV PROCESS; POLLING NETWORKS; THERMODYNAMIC LIMIT; MEAN-FIELD; PROPAGATION OF CHAOS."> <META NAME="dc.description" CONTENT="Delcoigne, Franck / Fayolle, Guy"> <META NAME="dc.publisher" CONTENT="INRIA"> <META NAME="dc.date" CONTENT="(SCHEME=ISO8601) 1998-04"> <META NAME="dc.relation" CONTENT="ftp://ftp.inria.fr/INRIA/publication/RR/RR-3398.ps.gz"> <META NAME="dc.relation" CONTENT="ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-3398.pdf"> <META NAME="dc.type" CONTENT="Text.TechReport"> <META NAME="dc.language" CONTENT="(SCHEME=ISO639-1) en"> <STYLE TYPE="text/css"> <!-- #Etiquette { visibility: visible } #UrFenetre  { position:absolute; width:171px; height:115px; z-index:8; left: 450px; top: 250px; visibility: hidden } #ProjetFenetre  { position:absolute; width:171px; height:115px; z-index:8; left: 450px; top: 250px; visibility: hidden } #PsFenetre  { position:absolute; width:171px; height:115px; z-index:8; left: 450px; top: 250px; visibility: hidden } #PdfFenetre  { position:absolute; width:171px; height:115px; z-index:8; left: 450px; top: 250px; visibility: hidden } #AuteurFenetre  { position:absolute; width:171px; height:115px; z-index:8; left: 450px; top: 250px; visibility: hidden }  --> </STYLE> <script language="JavaScript" src = "./showlayers.js"> </script> </HEAD><body> <table border="0" width="100%" cellpadding="6" cellspacing="0"> <!--  .............. logo INRIA .................. -->  <p style="margin-bottom: 16px"><a name="haut" href="http://www.inria.fr"><img src="../images/logo_inria.gif" width="102" height="34" alt="logo inria" border="0"></a></p> <h1><img src="../images/ti_dessus.gif" width="170" height="28" alt="-----------------------"><br>RR-3398 - Thermodynamic Limit and Propagation of Chaos in Polling Networks <BR>  <img src="../images/ti_dessous.gif" width="329" height="18" alt="-----------------------">  </h1> <BLOCKQUOTE> <FONT SIZE="4"><DIV ID="Etiquette"><A HREF="http://indexation.inria.fr:8080/cgi-bin/query?mss=RRRT%2Fsimple&pg=q&what=web&enc=iso88591&q=dc.creator%3DDelcoigne%2CFranck+url%3Ahttp%3A%2F%2Fwww.inria.fr%2Frrrt%2F&q1=dc.creator%3DDelcoigne%2CFranck&searchrrrt=Chercher"ONMOUSEOVER="MM_showHideLayers('document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','show','document.layers[\'AuteurFenetre\']','document.all[\'AuteurFenetre\']','show')" ONMOUSEOUT="MM_showHideLayers('document.layers[\'AuteurFenetre\']','document.all[\'AuteurFenetre\']','hide','document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','hide')"<!--div class=aut-->Delcoigne, Franck <!--/div--></A>  - <A HREF="http://indexation.inria.fr:8080/cgi-bin/query?mss=RRRT%2Fsimple&pg=q&what=web&enc=iso88591&q=dc.creator%3DFayolle%2CGuy+url%3Ahttp%3A%2F%2Fwww.inria.fr%2Frrrt%2F&q1=dc.creator%3DFayolle%2CGuy&searchrrrt=Chercher"ONMOUSEOVER="MM_showHideLayers('document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','show','document.layers[\'AuteurFenetre\']','document.all[\'AuteurFenetre\']','show')" ONMOUSEOUT="MM_showHideLayers('document.layers[\'AuteurFenetre\']','document.all[\'AuteurFenetre\']','hide','document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','hide')"><!--div class=aut-->Fayolle, Guy<!--/div--></A> </FONT></DIV><BR><DIV ID="AuteurFenetre"> <table border=6> <TR><TD bgcolor="#215E21">  <font color="#FFFFFF">Les rapports de cet auteur</font> </TD></TR></table></DIV> <DIV ID="Etiquette">Rapport de recherche de l'INRIA-<A HREF="/inria/organigramme/fiche_ur-rocq.fr.html" target="d"ONMOUSEOVER="MM_showHideLayers('document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','show','document.layers[\'UrFenetre\']','document.all[\'UrFenetre\']','show')" ONMOUSEOUT="MM_showHideLayers('document.layers[\'UrFenetre\']','document.all[\'UrFenetre\']','hide','document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','hide')"> Rocquencourt</A></DIV>  <DIV ID="UrFenetre"><table border=6><TR><TD bgcolor="#215E21">  <font color="#FFFFFF"> Page d'accueil de l'unit&eacute; de recherche</font> </TD></TR></table></DIV><P><P> <!--qqps--><DIV ID="Etiquette"><A HREF="ftp://ftp.inria.fr/INRIA/publication/RR/RR-3398.ps.gz"ONMOUSEOVER="MM_showHideLayers('document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','show','document.layers[\'PsFenetre\']','document.all[\'PsFenetre\']','show')" ONMOUSEOUT="MM_showHideLayers('document.layers[\'PsFenetre\']','document.all[\'PsFenetre\']','hide','document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','hide')">Fichier PostScript / <em>PostScript file</em></A></DIV> <DIV ID="PsFenetre"><table border=6> <TR><TD bgcolor="#215E21"> <font color="#FFFFFF">Fichier postscript du document :<BR>206  Ko </font></TD></TR></table></DIV><P> <!--qqpdf--><DIV ID="Etiquette"><A HREF="ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-3398.pdf"ONMOUSEOVER="MM_showHideLayers('document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','show','document.layers[\'PdfFenetre\']','document.all[\'PdfFenetre\']','show')" ONMOUSEOUT="MM_showHideLayers('document.layers[\'PdfFenetre\']','document.all[\'PdfFenetre\']','hide','document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','hide')">Fichier PDF / <em>PDF file</em></A></DIV> <DIV ID="PdfFenetre"><table border=6> <TR><TD bgcolor="#215E21"> <font color="#FFFFFF">Fichier PDF du document :<BR>404 Ko </font></TD></TR></table></DIV><P><DIV ID="Etiquette"><A HREF="../recherche/equipes/meval.fr.html" target="d"ONMOUSEOVER="MM_showHideLayers('document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','show','document.layers[\'ProjetFenetre\']','document.all[\'ProjetFenetre\']','show')" ONMOUSEOUT="MM_showHideLayers('document.layers[\'ProjetFenetre\']','document.all[\'ProjetFenetre\']','hide','document.layers[\'Etiquette2\']','document.all[\'Etiquette2\']','hide')">Equipe : MEVAL </A> - 45 pages - Avril 1998 - Document en anglais  </DIV> <DIV ID="ProjetFenetre"> <table border=6> <TR><TD bgcolor="#215E21">  <font color="#FFFFFF">Page d'accueil du projet</font> </TD></TR></table></DIV><P><em>Titre fran&ccedil;ais : Limite thermodynamique et propagation du chaos dans les r&eacute;seaux &agrave; scrutin </em><HR>Abstract :  {${\P\n,&cedil;N\geq 1 }$ is a sequence of standard polling networks, consisting of $N$ nodes attended by $V\n$ mobile servers. When a server arrives at a node $i$, he serves one of the waiting customers, if any, and then moves to node $j$ with probability $p_{ij}\n$. Customers arrive according to a Poisson process. Service requirements and switch-over times between nodes are independent exponentially distributed random variables. The behavior of $\P\n$ is analyzed in {\em thermodynamic limit}, i.e when both $N$ and $V\n$ tend to infinity, with $U\egaldef\lim_{N\rightarrow\infty}V\n/N,\ 0&lt;U&lt;\i- nfty$. First, ergodicity conditions are given. Then, combining the {\em mean-field} approximation approach together with weak convergence of Markov processes, the joint distribution [customers, vehicles] for an arbitrary finite number of nodes is explicitly characterized. In fact this distribution has a product form, which is the mathematical anologue of {\em the propagation of chaos}. One also computes the speed of convergence. In most of the study, $\P\n$ is a fully symmetrical network, but a generalization is carried out for systems provided with only {\em block-wise} symmetry. <P> <em>R&eacute;sum&eacute : {${\P\n,&cedil;N\geq 1 }$ d&eacute;signe une suite de r&eacute;seaux &agrave; scrutin ({\em polling}), form&eacute;s de $N$ noeuds et de $V\n$ serveurs mobiles. Lorsqu'un serveur arrive &agrave; une station $i$, il sert un &eacute;ventuel client en attente, puis se dirige vers le noeud $j$ avec probabilit&eacute; $p_{ij}\n$. Les arriv&eacute;es externes de clients forment un processus de Poisson. Les temps de service &agrave; chaque noeud, ainsi que les dur&eacute;es de d&eacute;placement inter-noeuds sont des variables al&eacute;atoires exponentielles, ind&eacute;pendantes. On analyse le comportement de $\P\n$ en {\em limite thermodynamique}, i.e. quand $N$ et $V\n$ tendent vers l'infini, avec $U\egaldef\lim_{N\rightarrow\infty}\fracV\n{N},\ &cedil; 0&lt;U&lt;\infty$. D'abord on donne les conditions d'ergodicit&eacute;. Ensuite, en combinant l'approche approximation {\em champ moyen}, utilis&eacute;e en physique statistique, avec la convergence faible pour les processus Markoviens, on caract&eacute;rise explicitement la distribution jointe [clients, v&eacute;hicules] pour un nombre fini quelconque de stations. Cette distribution est en fait un produit direct, qui refl&egrave;te la {\em propagation du chaos}. On donne &eacute;galement la vitesse de convergence. La plupart de l'&eacute;tude est r&eacute;alis&eacute;e pour des syst&egrave;mes sym&eacute;triques, mais une g&eacute;n&eacute;ralisation est donn&eacute;e dans le cas de sym&eacute;trie par bloc. </em><P><HR>KEY-WORDS : MARKOV PROCESS / POLLING NETWORKS / THERMODYNAMIC LIMIT / MEAN-FIELD / PROPAGATION OF CHAOS.<P>MOTS-CLES : PROCESSUS DE MARKOV / R&Eacute;SEAUX &Agrave; SCRUTIN / LIMITE THERMODYNAMIQUE / CHAMP MOYEN / PROPAGATION DU CHAOS<P> <HR> </BLOCKQUOTE> </BODY> </HTML> 
