When you are besieged there are four different options:

(1) Sally out and attack the besieging army – it takes some time to get your forces out thru the gate but the AI generally lets you accomplish that
(2) March another army to relieve the city – when you attack the city garrison acts as reinforcements
(3) Wait until the besieging army builds the capability to assault the walls and then hope that they do – I don’t think I have ever lost one of these battles unless I was severely outnumbered…of course I generally play as the English and longbowman stakes deployed all the right locations help
(4) Wait until you can no longer hold out and you are given the option sallying out or surrendering (I think)

Good luck