This paper considers a pair (F, τ ), where F is the flow of information and τ is a random time which might not be an F-stopping time. To this pair, we associate the new filtration G using the progressive enlargement of F with τ . For this setting governed with G, we analyze the optimal stopping problem in many aspects. Besides characterizing the existence of the solution to this problem in terms of F, we derive the mathematical structures of the value process of this control problem, and we single out the optimal stopping problem under F associated to it. These quantify deeply the impact of τ on the optimal stopping problem. As an application, we assume F is generated by a Brownian motion W , and we address the following linear reflected-backward-stochastic differential equations (RBSDE hereafter for short),   For this RBSDE, we focus on answering the following problems: a) What are the sufficient minimal conditions on the data (f, ξ, S, τ ) that guarantee the existence of the solution of the G-RBSDE in L p (p > 1)? b) How can we estimate the solution in norm using the triplet-data (f, ξ, S)? c) Is there an RBSDE under F that is intimately related to the current one and how their solutions are related to each other?