Pickup and delivery problems have numerous applications in practice such as parcel delivery and passenger transportation. In the dynamic variant of the problem, not all information is available in advance but is revealed during the planning process. Thus, it is crucial to anticipate future events in order to generate high-quality solutions. Previous work has shown that the use of waiting strategies has the potential to save costs and maximize service quality. We adapt various waiting heuristics to the pickup and delivery problem with time windows. Previous research has shown, that specialized waiting heuristics utilizing anticipatory knowledge potentially outperform general heuristics. Direct policy search based on evolutionary computation and a simulation model is proposed as a methodology to automatically specialize waiting strategies to different problem characteristics. Based on the strengths of the previously introduced waiting strategies, we propose a novel waiting heuristic that can utilize historical request information based on an intensity measure which does not require an additional data preprocessing step. The performance of the waiting heuristics is evaluated on a single set of benchmark instances containing various instance classes that differ in terms of spatial and temporal properties. The diverse set of benchmark instances is used to analyze the influence of spatial and temporal instance properties as well as the degree of dynamism to the potential savings that can be achieved by anticipatory waiting and the incorporation of knowledge about future requests.