  This paper attempts to study the optimal stopping time for semi-Markov
processes (SMPs) under the discount optimization criteria with unbounded cost
rates. In our work, we introduce an explicit construction of the equivalent
semi-Markov decision processes (SMDPs). The equivalence is embodied in the
value functions of SMPs and SMDPs, that is, every stopping time of SMPs can
induce a policy of SMDPs such that the value functions are equal, and vice
versa. The existence of the optimal stopping time of SMPs is proved by this
equivalence relation. Next, we give the optimality equation of the value
function and develop an effective iterative algorithm for computing it.
Moreover, we show that the optimal and {\epsilon}-optimal stopping time can be
characterized by the hitting time of the special sets. Finally, to illustrate
the validity of our results, an example of a maintenance system is presented in
the end.
