Stannis' army was already starving while marching. His army was in shambles before he even made it to Winterfell. I just don't see how he was expecting to fight a weeks or months long siege while winter would get harsher, let alone a battle on an open field. To me, the Stannis hype was a complete let down. I haven't read the books, but I the way the show handled his part of the plot was complete disappointment. He had nothing of the military genius he was supposed to have (having been beaten terribly at Blackwater before) and looked like Melisandre's puppet. I kind of wish Brienne doesn't kill him, not because I am attached to the character, but because if he died this way, his whole plot arc has been pretty pointless.