3 years ago

The Equilibrium States of Large Networks of Erlang Queues.

Davit Martirosyan, Philippe Robert

The equilibrium properties of allocation algorithms for networks with a large number of nodes with finite capacity are investigated. Every node is receiving a flow of requests and when a request arrives at a saturated node, i.e. a node whose capacity is fully utilized, an allocation algorithm may attempt to accommodate the request to another, non-saturated, node. For the algorithms considered, the re-routing comes at a price: either an extra-capacity is required or, when allocated, the sojourn time of the request in the network is increased. The paper analyzes the properties of the equilibrium points of the asymptotic associated dynamical system when the number of nodes gets large. At this occasion the classical model of {\em Gibbens, Hunt and Kelly} (1990) in this domain is revisited. The absence of known Lyapunov functions for the corresponding dynamical system complicates significantly the analysis. Several techniques are used: Analytic and scaling methods to identify the equilibrium points, probabilistic approaches, via coupling, to prove the stability of some of them and, finally, a criterion of exponential stability with the spectral gap of the corresponding linear system is derived. We show in particular that for a subset of parameters, the limiting stochastic model of these networks has multiple equilibrium points.

Publisher URL: http://arxiv.org/abs/1811.04763

DOI: arXiv:1811.04763v1

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