A zero-dimensional, or global, description of fuelling and exhaust processes is introduced to help assess techniques for reducing the density of gas, and its impact on plasma-density control, within the large vacuum tank of the MAST spherical tokamak. Model calculations for sample MAST discharges reveal their typical ion confinement time (≤50 ms) and contrasting fuelling efficiencies for outboard or inboard puffing. In steady state, the ratio between tank molecular and plasma densities is fixed irrespective of any sinks when gas is puffed from the vessel wall, but for fuelling at the plasma edge with moderate ionization efficiency, unsaturated boronization of MAST surfaces can yield ≈5 × lower gas density and excellent plasma pump-out. A 'two-chamber' adaptation of the global model with zones connected by variable conductances is also defined to address a possible in-vessel, cryopumped divertor. Exact limiting solutions reveal that for a constant plasma, steady-state gas density in the main torus generally rises for stronger divertor pumping when puffing into the tank, while it only decreases with pumping for fuelling at non-zero efficiency directly into the divertor plasma or 100% efficiency at any location. Full 'two-chamber' calculations for MAST indicate that the tank gas would be lowered principally through high divertor closure and fuelling efficiency, which combined with unsaturated boronization lead to > 10 × less steady gas density. Divertor cryopumping would be useful chiefly for plasma density control in long pulses or sustained exhaust in the absence of wall sinks. Global modelling offers a way similarly to examine fuelling behaviour in any tokamak device.