Compositional reasoning on hierarchical scheduling systems is a well-founded formal method that can construct schedulable and optimal system configurations in a compositional way. However, a compositional framework formulates the resource requirement of a component, called an interface, by assuming that a resource is always supplied by the parent components in the most pessimistic way. For this reason, the component interface demands more resources than the amount of resources that are really sufficient to satisfy sub-components. We provide two new supply bound functions which provides tighter bounds on the resource requirements of individual components. The tighter bounds are calculated by using more information about the scheduling system. We evaluate our new tighter bounds by using a model-based schedulability framework for hierarchical scheduling systems realized as Uppaal models. The timed models are checked using model checking tools Uppaal and Uppaal SMC, and we compare our results with the state of the art tool CARTS.