There is clearly a natural timescale to accrete matter around a black hole given by the balance of pressure and gravity, known as the Eddington luminosity. However, as I understand it, this is really a soft upper bound. In particular, an environment devoid of matter will clearly result in zero accretion. So, what about a nearly empty environment? In a rarified gas, I can't imagine forming a dense accretion disk, so it would seem that accretion should proceed as gas cools, forming a nearly isotropic halo around the black hole that is slowly accreted at the cooling rate.
In a slightly-less rarified scenario, I could imagine that an accretion disk does eventually form. It seems like the accretion rate should then ultimately depend on the properties of the disk. I would imagine that a high-viscosity disk should be accreted faster than a low-viscosity disk. However, this is apparently not how accretion works. The standard alpha disk model usually assumes that the viscosity determines the geometry of the disk, and that the mass accretion rate is an external parameter controlled by the environment. While it is clear that the environment should influence the accretion rate (see paragraph above), this picture cannot hold in general, since an environment dumping arbitrarily large amounts of mass onto the black hole would be limited by Eddington.
How do I reconcile these ideas? It would seem that the structure of the disk itself should somehow influence the mass accretion rate of the black hole. However, the environment should also play a role, since without an inflow of mass there would be no disk? The simplest picture I could imagine is that the accretion rate is something like $$ \dot M_{\rm BH} \approx \min(\dot M_{\rm environment},\dot M_{\rm Eddington})\,. $$
