Just a tiny nitpick, this is what fat people believe.
It's ackchyually up to two months if you have water. If you don't, it's about ten days. Happy hunger strikes!
Still, it would a lot less in a heavy stress + very active situation such as being inside a besieged city.
Having to run around, keep focus while fully equipped all while stressed with the possibility of enemy attack at any time increases your calories burned by a lot.
It also depends on how well fed you are and your nutrition level, both of which weren't too well in medieval times.
Upon research, starvation with water in current times indeed can take up to a month or more to actually kill, but still in a lot less it destroys your mood (morale very low) it causes weakness and boneloss (reduced health/already starts injured and reduced attributes), loss of focus (reduced accuracy) and after a while, severe weakness in which there would be no possibility to fight at all (wounded state).
Up to a month for someone not active, as I said, a militia and garrison during a siege would be very active, which reduces the time needed to closer to a week, without food. And this has been show in several periods of history in which an army had their supply line destroyed and blocked. 1 week without any food at all is more than enough for an army to be unable to fight and for a town to fall due to not being able to defend itself anymore at most 10 days.
It still would be a progressive loss of troops, starting low and raising with times, but maybe instead of *2 in the 20, 40, 80, 160,... progression I first suggested, a *1,5 or *1,2 would be closer
q(1,5) = 20, 30, 45, 67.5, 101.2
q(1,2) = 20, 24, 28.8, 34,5, 41,4
At least it would be growing progressively and in a few days you'd clearly see huge losses from one day to the next by people giving up or too weakened to fight.
But currently, I held a town without food for quite literally a month (it was a rebel owned settlement, so there was no army coming to help) and the number of defenders only reduced from 400 to 200 or 170 something, somewhere still in the hundreds, after 30 days without any food at all.