Optimal stopping time on discounted semi-Markov processes

TL;DR

This paper investigates the optimal stopping problem for discounted semi-Markov processes with unbounded costs, establishing an equivalence with semi-Markov decision processes and providing algorithms for computing optimal policies.

Contribution

It introduces an explicit construction linking SMPs and SMDPs, proving the existence of optimal stopping times and developing an iterative algorithm for their computation.

Findings

Established the equivalence between SMPs and SMDPs in terms of value functions

Proved the existence of optimal stopping times for SMPs

Developed an iterative algorithm for computing optimal policies

Abstract

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.