Without speaking too generally, sieges were a calculated conflict that would be maintained so long as the attacker could justify the expense and avoid allies of the besieged who were trying to break the siege through pitched battle. However, before the advent of effective artillery rarely did a siege end without some sort of surrender or agreement between the two parties. A rough estimate would be 8-12 months, but Edward III lifted a siege after 17 days during the Hundred Years War once.