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 Markov jump processes are analyzed under a thermodynamic limit regime, that is, when the networks have some symmetry properties and when the number of nodes goes to infinity. An intriguing stability property is proved: under some conditions on the parameters, it is shown that, in the limit, several stable 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. Networks with a unique equilibrium point are also presented.

This work addresses the problem of estimating the distributions of packet flow sizes and durations under several methods of sampling packets. Two approaches, one based on inversion and the other on asymptotics, are considered. For the duration distribution, in particular, both approaches require modeling the structure of flows, with the duration distribution being characterized in terms of the IATs (interarrival times between packets) and size distributions of a flow. The inversion of the flow IAT distribution from sampled flow quantities, along with the inversion of the flow size distribution (already used in the literature) allows estimating the flow duration distribution. Motivated by the limitations of the inversion approach in estimating the distribution tails for some sampling methods, an asymptotic approach is developed to estimate directly the distribution tails of flow durations and sizes from sampled quantities. The adequacy of both approaches to estimate the flow distributions is checked against two real Internet traces.

This paper investigates the distributions of triangle counts per vertex and edge, as a means for network description, analysis, model building, and other tasks. The main interest is in estimating these distributions through sampling, especially for large networks. A novel sampling method tailored for the estimation analysis is proposed, with three sampling designs motivated by several network access scenarios. An estimation method based on inversion and an asymptotic method are developed to recover the entire distribution. A single method to estimate the distribution using multiple samples is also considered. Algorithms are presented to sample the network under the various access scenarios. Finally, the estimation methods on synthetic and real-world networks are evaluated in a data study.

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, in general, have reversibility properties, so the explicit expression of their invariant distribution is not known. Kelly’s limiting 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 limit, the associated rescaled Markov process converges to a deterministic dynamical system with a unique equilibrium point characterized by a nonstandard fixed point equation.