  Consider $N$ stations interconnected with links, each of capacity $K$,
forming a complete graph. Calls arrive to each link at rate $\lambda$ and
depart at rate $1$. If a call arrives to a link $x y$, connecting stations $x$
and $y$, which is at capacity, then a third station $z$ is chosen uniformly at
random and the call is attempted to be routed via $z$: if both links $x z$ and
$z y$ have spare capacity, then the call is held simultaneously on these two;
otherwise the call is lost.
  We analyse an approximation of this model. We show rigorously that there are
three phases according to the traffic intensity $\alpha := \lambda/K$: for
$\alpha \in (0,\alpha_c) \cup (1,\infty)$, the system has mixing time
logarithmic in the number of links $n := \binom N2$; for $\alpha \in
(\alpha_c,1)$ the system has mixing time exponential in $n$, the number of
links. Here $\alpha_c := \tfrac13 (5 \sqrt{10} - 13) \approx 0.937$ is an
explicit critical threshold with a simple interpretation. We also consider
allowing multiple rerouting attempts. This has little effect on the overall
behaviour; it does not remove the metastability phase.
  Finally, we add trunk reservation: in this, some number $\sigma$ of circuits
are reserved; a rerouting attempt is only accepted if at least $\sigma+1$
circuits are available. We show that if $\sigma$ is chosen sufficiently large,
depending only on $\alpha$, not $K$ or $n$, then the metastability phase is
removed.
