The attackers would suffer starvation because they are in a foreign land and far from their supply lines.
Knight_Yellow:


Also, they need to keep a large amount of men watching the castle so the defenders don't sally out and kill them so they can't have all that many men out foraging. Plus by sitting in one place for such a long time they use up all the food in the area. Medieval supply lines weren't exactly fantastic - they pretty much needed to get local food supplies to survive in enemy territory.

When armies are marching through the enemy countryside they loot and raid farms and villages for food supplies but in sieges the troops can't stray far from the castle so there isn't that much food.

Also, about combat casualties

If the attackers aren't moving into bowshot to shoot at the defenders and cause casualties, then why are the defenders dying at a steady rate? If the cause was only starvation, I figure they would die in small numbers at first and then as time went on they would start dropping considerably.