To number them and determine the ties, you begin by sorting up by time, and then you unpropagate, so that when there are ties the first one remains, but the succeeding matching times become empty.
You then do a FormulaFill in the place field, with seq() going there if the time is not blank, and zeroblank(0) going there if it is. Then you propagate so that the place awarded to the first of the ties is propagated to the others.
Then you go back to the time field and propagate it to restore the times that were made blank.