I'm pretty sure I'm just having trouble finding a quote, but I can't figure out what season it's supposed to be at the start of the Siege of Luthadel. It seems like the siege ends in the winter, due to the snow when the koloss attack, but I can't figure out either exactly how long the siege lasted or what time of year it was throughout.
It was fall when the siege started, and winter by the time it ended.
Relevant quote from The Well of Ascension, as they discuss the first two incoming armies (the third not showing till later) and propose the plan to play the two off each other:
“Not to mention our food problems,” Dockson said. “What you propose would take time, Your Majesty. Time during which we’d be under siege, our supplies dwindling. It’s autumn right now. Winter will soon be upon us.”
Winter ending of the seige can be easily concluded by the snow on the battlefield, which even Elend has to deal with on his trip north towards Terris.