I think AnnO has got the correct grammer here without a doubt, but a more correct translation would be " had begun to reign,at sometime prior to this staement". In writing it as just has begun to reign implies that it has only happen at that imediate time, where as the tense indicates a time in the past. It is more simply underastood when the tense is qualified with a date and time but in this case it is not.
It is interesting to note that JW.Org changed the meaning of Luke 10:18 " At that he said to them "I began to behold Satan already fallen like lightening from heaven" (old NWT) and " ........I see Satan already fallen......from heaven." (new NWT). Both of these translations are wrong for the same reason and in this case the action of falling from heaven is complete not ongoing . Jesus is saying that at a time in the past which is unspecified he had been present at event when Satan fell from heaven and physically saw it happen. This statement unfortunately places the events of Rev 12:9 prior to this time and does not agree with the 1914 doctrine, neither does he say that he was doing the throwing/hurling, he was only spectating the event.
BB