Danielle Tibi

Stochastic networks with multiple stable points

This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated (non-reversible) Markov jump processes are analyzed under a thermodynamic limit regime, i.e. when the networks have some symmetry properties and when the number of nodes goes to infinity. A metastability property is proved: under some conditions on the parameters, it is shown that, in the limit, several equilibrium points coexist for the empirical distribution. The key ingredient of the proof of this property is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their local minima and saddle points. Cases with a unique equilibrium point are also presented.

Analysis of Loss Networks with Routing

This paper analyzes stochastic networks consisting of finite capacity nodes with different classes of requests which move according to some routing policy. The Markov processes describing these networks do not have, in general, reversibility properties so that the explicit expression of their invariant distribution is not known. A heavy traffic limit regime is considered: the arrival rates of calls as well as the capacities of the nodes are proportional to a factor going to infinity. It is proved that, in the limit, the associated rescaled Markov process converges to a deterministic dynamical system with a unique equilibrium point characterized by a non-standard fixed point equation.

Metastability of CDMA cellular systems

In this paper, it is shown that the coexistence of a variety of different traffics in third generation cellular networks may lead to a very undesirable behavior of the whole network: a metastability property. When this property holds, the state of the network fluctuates on a very long time scale between different set of states. These long oscillations of the network make impossible to predict the average performances of some of the key characteristics of the connections, such as the handoff blocking rate or the probability of call blocking. As a consequence, the quality of service provided by such a network can be guaranteed only by, sometimes poor, lower bounds. Experiments of a UMTS network with this behavior are presented and the analysis of a corresponding simplified mathematical model is developed. The practical implications in the design of radio resource management for CDMA cellular networks are discussed.

Equivalence of ensembles for large vehicle-sharing models

For a class of large closed Jackson networks submitted to capacity constraints, asymptotic independence of the nodes in normal traffic phase is proved at stationarity under mild assumptions, using a Local Limit Theorem. The limiting distributions of the queues are explicit. In the Statistical Mechanics terminology, the equivalence of ensembles - canonical and grand canonical - is proved for specific marginals. The framework includes the case of networks with two types of nodes: single server/finite capacity nodes and infinite servers/infinite capacity nodes, that can be taken as basic models for bike-sharing systems. The effect of local saturation is modeled by generalized blocking and rerouting procedures, under which the stationary state is proved to have product-form. The grand canonical approximation can then be used for adjusting the total number of bikes and the capacities of the stations to the expected demand.

Analysis of a network model

In this paper we study via a queueing analogy a stochastic model of a neural network of Purkinje cells proposed by (Cottrell 1991). In this model only the inhibitory interaction between the cells is considered. We analyze the stability properties of networks whose graph is complete N-partite. It is shown that if the parameters of the model are below some critical values, then the network converges to unique equilibrium. In this case, the invariant measure of the inhibition states is explicited as well as the distribution of the duration of the inter-spikes of a given cell. If this stability condition is not satisfied, the state of the network converges to some asymptotic state depending on its initial state. In this case, the set of possible asymptotic states is given.

Study of a stochastic model for mobile networks

Recent wireless technologies have triggered interest in a new class of stochastic networks, called mobile networks in the technical literature. In contrast with Jackson networks where users move upon completion of service at some node, in these mobile networks, transitions of customers within the network occur independently of the service received. Moreover, at any given time, each node capacity is divided between the users present, whose service rate thus depends on the capacity and on the state of occupancy of the node. Once his initial service requirement has been fulfilled, a customer definitively leaves the network. In [3], complex capacity sharing policies are considered, but in the simplest setting, which will be of interest to us, nodes implement the Processor-Sharing discipline by dividing their capacity equally between all the users present. Previous works [3, 6] have mainly focused on determining the stability region of such networks, and it has been commonly observed that the users’ mobility represents an opportunity for the network to increase this region. Indeed, because of their mobility, users offer a diversity of channel conditions to the base stations (in charge of allocating the resources of the nodes), thus allowing them to select the users in the most favorable state. Such a scheduling strategy is sometimes referred to as an opportunistic scheduling strategy, see [2] and the references therein for more details.