Thermodynamic Limit and Propagation of Chaos in Polling Networks

Projet MEVAL{{\P\n,¸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,\

Large Deviations Problems for Star Networks: the Min Policy Part I: Finite Time

Projet MEVALIn this paper, we prove a sample path large deviation principle for a rescaled process $n^{-1}Q_{nt}$, where $Q_t$ represents the joint number of connections at time $t$ in a star network where the bandwidth is shared between customers according to the so-called min policy. The rate function is computed explicitly. One of the main steps consists in deriving large deviation bounds for an empirical generator constructed from the join number of customers and arrivals on each route. The rest of the analysis relies on a suitable change of measure together with a localization procedure

Large deviations for a class of Markov processes modelling communication networks

In this paper, we prove a sample path large deviation principle (LDP) for a rescaled process n^-1Q_nt, where Q_t is a multi-dimensional birth and death process describing the evolution of a communication network. In this setting, Q_t is the join number of documents on the set of routes at time t. Documents to be transferred arrive on route r as a Poisson process with rate _r and are transferred at rate _r_r(x) where x represents the state of the network, _r^-1 is the mean size of documents on route r and _r(x) is the bandwidth allocated to route r. We describe a set of assumptions over the allocation under which the LDP holds. Since we want the «classical» allocatio- ns to verify these assumptions, the difficulty is to deal with weak properties- . For example, _r(x) is assumed to be continuous on the set _r=x:x_r>0 but may be discontinuous elsewhere. Several examples are provided including the max-min-fairness allocation, a classical one in the context of data networks. Since the main object to work with is the local rate function, a great care has been devoted to its expression and its properties. It is expressed as the solution of a convex program from which many useful properties are derived. We believe that this kind of expression allows numerical computations

On Polling Systems where Servers wait for Customers

Projet MEVALIn this paper, a particular polling system with $N$ queues and $V$ servers is analyzed. Whenever a server visits an empty queue, it waits for the next customer to come to this queue. A customer chooses his destination according to a routing matrix $P$. The model originates from specific problems arising in transportation networks. A global classification of the process describing the system is given under general assumptions. It is shown that this process can only be {\em transient} or {\em null recurrent}. In addition, a detailed classification of each node, together with limit laws (after proper time-scaling) are obtained. The method of analysis rel= ies on the central limit theorem and a coupling with a reference system in which transportation times are identically zero

Thermodynamic limit and propagation of chaos in polling networks

Theme 1 - Reseaux et systemes. Projet MevalSIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1998 n.3398 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Large deviations for polling systems

Theme 1 - Reseaux et systemes. Projet MEVALAvailable from INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.2000 n.3892 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc

On polling systems where servers wait for customers

Theme 1b - Reseaux et systemes. Projet MEVALSIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1996 n.3058 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc