In this paper, we show, for the case of good filtration of ideals, that  the relation between the invariants characterizing the homogeneous vanishing of the local cohomology of the Rees algebra and the associated graded ring behaves similarly to the case of adic-filtration. Moreover, we obtain, in certain cases, inequalities (possibly the equality) between the Castelnuovo-Mumford regularity of the fiber cone  and the regularity of the associated graded ring  as  is a good filtration. For instance, if  is the analytic spread of  and  then we show  

Castelnuovo-Mumford regularity, fiber cone, associated graded ring, Rees algebra, good filtration, reduction number.