We don't have limited run tv shows on US network tv.
They generally have episodes in the can before premiere (3-6), then get greenlit for a higher order or they burn off the filmed episodes if the ratings are low.
What you are describing if translated to the US is a miniseries and they do not get put on the sched. If its big budget enough, they run during sweeps like with The Odyssey. If its lower budget or a failed network show they turn into a movie, they get run during a slower period or go STVĀ (The Storm maybe? or Cruel Intentions 2 def). Minis are more prominent on cable or the premiums.
Prime Suspect doesn't fit your description and there is no advertising/awareness of the British show here. Its on Thurs. 10-11pm, getting promo time (less than Whitney), and assuming short or long longterm without seeing ratings.....not even close to enough information to gauge that.