Originally Posted by SilverTauron
This is static thinking.
At some point you WILL exhaust your resources. It may take weeks or months, but the bad guys have the time. They can set up camp and re-supply as needed. You as the defender don't have that luxury. Once you run out of stores , game over.
However, since such fixed positions with the stores necessary to weather a siege of some length are the source
of the supplies of the besiegers, there's no reason to assume a long siege. Easier to bypass and come back later.
Historically, sieges were raised when
* other requirements caused the besieging forces to move
* a relief force was near enough to threaten to or actually engage the besiegers (probably N/A here)
* sickness weakened the besiegers
* besiegers were unable to forage for enough supplies to stay
* besiegers were unable to cut off the besieged from resupply, and finally gave up
* a traitor opens a gate and the redoubt falls
* sickness in the strong point reduced the defenders beyond the ability to defend
* lack of supplies weakens the defenders beyond the ability to defend, as suggested here
* a message is received by the defenders ordering them to surrender (probably N/A here)
* a military or negotiated resolution is reached at a higher level, causing hostilities to cease. (probably N/A here)
ETA - 1 more, exemplified by Masada - garrison suicides rather than yield.
(And I did omit 'attackers win by assault' - climb the walls, break the gates, undermine the walls/towers, etc - and kill some or all of the defenders. No more siege after that.)
I don't recall reading of any where the besieged managed to sally and break the siege, but that's a possibility. As you point out, tough for a family of 4 to do against a larger force.