  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.
