For a simulated annealing process $X_t$ on S with transition
rates $q_{ij}(t) = p_{ij} \exp (-(U(i, j))/T(t))$ where $i, j \epsilon S$ and
$T(t) \downarrow 0$ in a suitable way, we study the exit distribution
$P_{t,i}(X_{\tau} = a)$ and mean exit time $E_{t,i}(\tau)$ of $X_t$ from a
cycle c as $t \to \infty$. A cycle is a particular subset of S
whose precise definition will be given in Section 1. Here $\tau$ is the exit
time of the process from c containing i and a is an
arbitrary state not in c. We consider the differential (backward)
equation of $P_{t,i}(X_{\tau} = a)$ and $E_{t,i}(\tau)$ and show that
$\lim_{t\to\infty}P_{t,i}(X_{\tau} = a)/\exp (-U(c, a) - T(t))$ and
$\lim_{t\to\infty E_{t,i}(\tau)/\exp(V(c)/T(t))$ exist and are independent of
$i \epsilon c$. The constant $(U(c, a))$ is usually referred to as the cost
from c to a and $V(c), (\leq U(c, a))$ is the minimal cost coming
out of c. We also obtain estimates of $|P_{t,i}(X_{\tau} = a) -
P_{t,j}(X_{\tau} = a)|$ and $|E_{t,i}(\tau)|$ for $i \not= j$ as $t \to
\infty$. As an application, we shall show that similar results hold for the
family of Markov processes with transition rates $q_{ij}^{\varepsilon} = p_{ij}
\exp(-U(i, j)/\varepsilon)$ where $\varepsilon > 0$ is small.