The demand for Z would be greater than the supply (fixed by assumption) if the cost of this time was less than the equilibrium price of Z before the price control.
The scrambling by households for the limited supply would increase the time required to get a unit of Z, and thus its cost.
Both would continue to increase until the average cost of time tended to the equilibrium price before price control.
At that point equilibrium would be achieved because the supply and demand for Z would be equal.