I'd also be interested in knowing why we don't seem to hear about sieges in 20th century or modern warfare any more. Starving your enemy out of a city seems like a very effective strategy.
The Siege of Leningrad in WWII lasted 872 days, and was one of the most destructive sieges in history.
The Germans and the Finns besieged Leningrad. The Russians defended. The siege was rarely complete, as supplies (in small numbers) could still get in across Lake Ladoga most of the time.
Up to 1.5m Russian soldiers and civilians died in the siege. Famine and disease were the main killers. 1.4m civilians were managed to be evacuated.