An attacking army surrounds a well-built castle that it cannot storm, and no relief force is coming. Which siege method would commonly be used, and what did it depend on?

Explanation

Cutting a garrison off from food and water shifted the contest from the walls to the storerooms, and it decided a siege slowly rather than quickly, since the outcome turned on supplies running low over weeks or months instead of on one day of fighting. A single rush at the gate fails as the general method because a castle entrance was designed precisely to make such an attempt costly, which is why blockade was reached for when storming was not realistic.