For a large class (satisfying a Perron-Frobenius property) of simulated annealing processes with an absorbing state a and arbitrary cost function, it is shown that there exist constants h(i) greater-than-or-equal-to 0, beta(ij) > 0 and delta > 0, N greater-than-or-equal-to 0, independent of the starting points such that, for nonabsorbing states i and j, (GRAPHICS) Here, lambda(t) = exp(-1/T(t)), T(t) --> 0, is the temperature function. As an application, the asymptotic behavior of the expected time of hitting a state (in particular, a global minimum) of a simulated annealing process without absorbing states can be determined.