With over 600,000 male Israelites required to offer sacrifices at the Tabernacle, how many males came at one time? This seems to comprise an enormous logistical difficulty.
I can think of 2 cases when everybody had to bring a sacrifice: Pessach and Festivals.
For Korban Pessach they had more than 1 person per animal - usually a family (or more) per animal. But even then they divided the crowd into 3 - and they sacrificed the Pessach in 3 groups.
They had an entire procedure as described in the Mishna in Pesachim to make the process efficient.
The Chagigah and Re'iya that had to be brought at each of the 3 festivals, one had 7 days (8 on Sukkoth) to bring them - so that would have alleviated the rush somewhat.
There's a Mishna that says that the Temple Courtyard was filled to capacity before dawn with people lining up to bring their sacrifices - so I guess it was simply done on a first-come-first-serve basis.