Ideas for volunteer schedule for non-profit event

I'm helping with the website for the host committee for AA's 75th anniversary in San Antonio in 2010. The event will require around 5000 volunteers to accommodate the 60,000 recovering drunks hitting the Alamo City next July. The organizers would like for volunteers to be able to sign up for time slots for different jobs.
For example, 60 volunteers are needed for the Tuesday 1pm-3pm Hotel Lobby Greeters slot. Once 60 volunteers sign up, that slot should not be available any more, unless someone drops it.

I've been looking for a way to make this happen and have not been able to come up with a simple solution. Should it be that each time slot is a node? It seems like a common issue so I'm guessing there is a contributed module that should cover this. I have not found one.

I really need some suggestions as I try to wrap my head around this problem.
Here is a copy of how the schedule might look and all the different slots and required volunteers per slot:
http://aa75.org/test-schedule

Any ideas will help...

Thanks,

Daniel