Equilibrium would take different forms depending on the method of rationing.
With a literal " first come first served " system the size of the queue (say outside the bakery or in the doctor's office) would grow until the expected cost of standing in line discouraged any excess demand;' with the figurative queues of appointment systems, the " waiting " time (say to see a play) would grow until demand was sufficiently curtailed. If the system or rationing was less formal, as in the labour market during recessions, the expected time required to ferret out a scarce job would grow until the demand for jobs was curtailed to the limited supply