The Equilibrium States of Large Networks of Erlang Queues
Davit Martirosyan, Philippe Robert
TL;DR
This work analyzes the equilibrium structure and stability of large networks of Erlang queues under two rerouting schemes, using mean-field limits to derive nonlinear dynamical systems. It combines analytic, scaling, and probabilistic techniques to identify conditions under which multiple equilibria exist, and to establish scenarios of exponential stability via spectral-gap criteria and coupling arguments. The RIST and DAR algorithms are shown to exhibit regimes with multiple equilibria, and the authors provide both local stability conditions and global-like bounds for saturated regimes, linking fixed-point analysis to McKean–Vlasov limits and non-linear M/M/1 queue behavior. The results offer rigorous insights into the design trade-offs for rerouting policies, highlighting when reallocation can sustain stability and when saturation dynamics dominate performance.
Abstract
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 re-allocate the request to a non-saturated node. For the algorithms considered, the re-allocation comes at a price: either an extra-capacity is required in the system or the processing time of a re-allocated request 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. We identify the subset of parameters for which the limiting stochastic model of these networks has multiple equilibrium points. Probabilistic approaches, like coupling, are used to prove the stability of some of them. A criterion of exponential stability with the spectral gap of the associated linear operator of equilibrium points is also obtained.