Essential Faces and Stability Conditions of Multiclass Networks with Priorities

It is now well-known that multiclass networks may be unstable even under the \usual conditions" of stability (when the loads are less than one at all queues), but the proofs of transience (in the Markovian case) generally require a complex work based on the dynamics of an associated \\uid model". Here we develop a sample-path argument introduced in a previous paper, which provides new ergodicity conditions for networks ruled by priorities; when one of these conditions is violated, the network diverges at linear speed. Our approach is based on the identiication of the essential faces, which are the sets of classes that can be simultaneously occupied in stationary regime. A graph is associated with the network, the existence of unessential faces being equivalent to the presence of cycles in this graph, in which case the usual conditions are not suucient conditions of stability. As a by-product of our results, we recover the stability conditions and complete the analysis of two exemplary models, the Rybko-Stolyar network and the Lu-Kumar network. Faces essentielles et conditions de stabilit e des r eseaux multiclasses avec priorit es R esum e : On sait maintenant que les r eseaux multiclasses peuvent ^ etre instables sous les \conditions habituelles" de stabilit e (c'est-a-dire quand la charge est inf erieure a 1 a chaque le), mais prouver que le mod ele est transient (dans le cas markovien) exige g en eralement un travail fastidieux fond e sur l' etude de la dynamique du \mod ele uide" associ e. Nous d eveloppons ici un argument purement trajectoriel introduit dans un article pr ec edent, lequel fournit de nouvelles conditions d'ergodicit e pour les r eseaux r egis par des priorit es ; si l'une de ces conditions est viol ee, le r eseau diverge a vitesse lin eaire. Notre approche est fond ee sur l'identiication des faces essentielles, c'est-a-dire des ensembles de classes pouvant ^ etre occup ees simultan ement en r egime stationnaire. L'existence de faces non essentielles equivaut a la pr esence de cycles dans un graphe associ e au r eseau, auquel cas les conditions habituelles ne sont pas suusantes pour rendre le r eseau stable. Comme corollaire de nos r esultats, nous retrouvons les conditions de stabilit e de deux mod eles de r ef erence, le r eseau de Rybko-Stolyar et celui de Lu-Kumar. Mots-cl es : R eseaux de les d'attente multiclasses, …