The revised Schedule of Classes will make it easy to list all slots. Since the Registrar controls x classrooms and the new Schedule will have y time periods per week, there will be xy slots for Registrar-controlled classrooms.
People sign up for slots first come, first served.
A sophisticated search engine could help department administrators and faculty members locates slots with characteristics they specify, and would indicate whether the slots had been reserved. However, conflicts inevitably will arise, and first come-first served is not a good principle for resolving them. First come-first served has the advantage of encouraging departments to finalize their teaching schedules early, but it also allows them to hoard slots by signing up with courses that may not be taught. There is no reason to believe that the first person to sign up always will be the best user of a classroom, and no one gets information about alternative uses. Suppose that each department each semester has a fixed allotment of points.