BRDF physical constraints

I have disscussed a lot about how to implememt and sampling the traditional BRDF models in a path tracing render system which is physically based. But recently I've been tired of the trade off between physics and production. Because of the existance of models like Blinn-phong or phong, it is very hard to adjust a physically plausible result in the shaders. I strongly recommanded to replace these old models with microfacet. But here I still need to discuss how to deduce the constraints.

1. Definitions

We take phong for exemple:

2.BRDF albedo

We made it be the total hemispherical reflection, the intergration is:

when we have the perpendicularly incoming direction, the later intergration reach its maximum. Which in this case:

We can transform the later intergration to :

We always have:

This intergration has a upper limit .

If we regard the later intergration as 1. We will get the result kd + ks is the BRDF albedo, so when using the phong, we must keep kd+ks<1