I don't think that it lets you have longer bash lag then warriors without it but rather that you will miss fewer bashes and be in the high end of the possible lag range more often.
so its always 1-4 rounds lag but a normal warriour would average a 2,5 and with greeting you would average 3, I don't think the bonus is super huge but hey all you really need is that extra pulse that lets you fire a second bash before they get a command. if your bashing every 2 rounds it doesn't matter if your lagging them 2.1 or 4 since your just replacing any extra lag with the new bash.
