All 3 Uses

besiege
in
1776, by McCullough
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The miseries of the troops still besieged at Boston, and of those Americans loyal to the King who, fearing for their lives, had abandoned everything to find refuge in the town, were also described in letters published in the London papers or in correspondence to friends and relatives in London.†
p. 8.5

besieged = surrounded and cut-off supplies; or overwhelmed
Supplying the besieged city by sea had become increasingly difficult because of winter storms and American privateers.†
p. 60.9 *
At Boston, Washington had benefited greatly from a steady supply of valuable intelligence coming out of the besieged town, while Howe had known little or nothing of Washington's strengths or intentions.†
p. 119.8

Definitions:

(1) (besiege)
to surround a place with armed forces and cut off supplies or attack; or more generally, to overwhelm someone with many requests, complaints, or attention, as if surrounding them and not letting up
(2) (meaning too rare to warrant focus)