1. A method of facilitating processing in a processing environment, said method comprising: responsive to occurrence of a page fault associated with a data request of a task currently executing, suspending execution of the task, indicating the task as being currently suspended for page fault processing, and placing the suspended task at an end of a dispatch structure of tasks ready for execution by a processor; performing, by one or more components of the processing environment, a page fault operation for the suspended task, wherein the suspended task remains on the dispatch structure during performance of the page fault operation for the suspended task; based on retrieving data to satisfy the page fault, indicating completion of the page fault operation by setting, by a first processor of the processing environment, a completion indicator associated with the suspended task, the completion indicator indicating that execution of the suspended task may be resumed based on identifying the suspended task as potentially next for dispatching from the dispatch structure; and performing dispatching of tasks from the dispatch structure, wherein at least one suspended task for which an operation has yet to be completed remains on the dispatch structure while at least one other task on the dispatch structure is selected for potentially resuming execution thereof, and wherein the performing comprises: responsive to identifying, on the dispatch structure, the suspended task as potentially next for execution, checking prior to dispatching of the suspended task, the completion indicator associated with the suspended task to determine whether the completion indicator is set indicating completion of the page fault operation, and, based on the checking indicating completion of the page fault operation, dispatching the suspended task from the dispatch structure to a second processor of the processing environment, to resume execution of the suspended task, wherein the first processor may be the same or different from the second processor.