Analysis of loss networks with routing

TL;DR

This paper studies stochastic loss networks with routing, analyzing their behavior in large-scale limits where the network's stochastic process converges to a deterministic system with a unique equilibrium.

Contribution

It introduces a limiting regime analysis for non-reversible loss networks, deriving the convergence to a deterministic dynamical system with a novel fixed point characterization.

Findings

Rescaled Markov processes converge to a deterministic system

Unique equilibrium point identified via a nonstandard fixed point equation

Provides insights into large-scale network behavior

Abstract

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.