  We use a method of Luczak (arXiv:1212.3231) to investigate the equilibrium
distribution of a dynamic routing model on a network. In this model, there are
$n$ nodes, each pair joined by a link of capacity $C$. For each pair of nodes,
calls arrive for this pair of endpoints as a Poisson process with rate
$\lambda$. A call for endpoints $\{u,v\}$ is routed directly onto the link
between the two nodes if there is spare capacity; otherwise $d$ two-link paths
between $u$ and $v$ are considered, and the call is routed along a path with
lowest maximum load, if possible. The duration of each call is an exponential
random variable with unit mean. In the case $d=1$, it was suggested by Gibbens,
Hunt and Kelly in 1990 that the equilibrium of this process is related to the
fixed points of a certain equation. We show that this is indeed the case, for
every $d \ge 1$, provided the arrival rate $\lambda$ is either sufficiently
small or sufficiently large. In either regime, we show that the equation has a
unique fixed point, and that, in equilibrium, for each $j$, the proportion of
links at each node with load $j$ is strongly concentrated around the $j$th
coordinate of the fixed point.
