Appointments assigned by times to staff?

I’m pretty adept with WordPress and I’m trying to find the best plugin or solution for a university web page. I need a standard booking/appointment plugin (by half hour increments) but I don’t want clients to be able to pick out who they want an appointment with. I just want them to pick a time and have the back-end send an appointment reminder to the user as well as a member of university staff scheduled/assigned to those times. Ideally, I could go into the plugin, assign staff (assume they’ll be admins on WordPress) to cover certain times of random days out of the week that users will makes appointments for. NOTE: times could vary ridiculously (e.g. Sue says she can cover appointments 8:30-12, 1-3:30, and 4:30-5). They’re all trained to give the same kind of feedback; I don’t want people thinking it’s advantageous to have an appointment with certain individual over another when that’s not true. Thanks for your time!

Bottom Line: An appointments plugin organized by shifts.

The page I need help with: [log in to see the link]