“Besiege” uses the first meaning of “be-“, to surround physically.  Surround the enemy, block the flow of food in to them, and wait for surrender.

• 15.058 I declare the Mountain besieged.
• 15.059 and being besieged inside it was not at all to his taste.
• 17.034 so that the besiegers
• 17.038 of the besiegers

“besiege, v.” OED Online. Oxford University Press, March 2015. Web. 7 May 2015.