POTENTIAL CONFLICT BTWN 2 ACR WHEN THE FLC OF A B737-900 GETS BEHIND THE PWR CURVE ON THEIR MISSED APCH AND A SECOND APCH AT IAH; TX.

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WE WERE FLYING FROM LOS ANGELES TO HOUSTON (IAH). THE WX AT THE ARPT WAS RPTED TO BE 500 FT OVCST AND VISIBILITY OF 10 MI. HOUSTON APCH CTL TOLD US TO PLAN FOR AN ILS RWY 27. MY FO WAS FLYING THIS LEG TO HOUSTON. DURING THE RADAR VECTORS FOR THE APCH WE WERE LEVEL AT 4000 FT. NEXT TURN TO THE LOOSE LEG WE WERE CLRED DIRECT TO THE FINAL APCH FIX; DSND TO 2000 FT AND CLRED FOR THE APCH. MY FO DID NOT BEGIN TO DSND RIGHT A WAY AND WE ENDED UP HIGH ON THIS APCH. THIS ALSO LED TO AN UNSTABLE APCH AND LNDG. I ORDERED HIM TO EXECUTE A MISSED APCH. THE TWR GAVE US A HEADING AND ALT TO CLB UP TO. THE NEXT CTLR TOLD US TO CLB TO 4000 FT. THIS EVENING HOUSTON OPERATED BOTH ILS RWY 26 AND ILS RWY 27 APCHS. THE NEXT HEADING WE RECEIVED WAS TO INTERCEPT THE LOC FOR THE ILS RWY 26. THAT SURPRISED US AND WE WERE NOT PREPARED FOR THAT APCH. ALL OUR RADIOS AND NAVAIDS WERE SET FOR ILS RWY 27. WE ALSO HAD A STRONG XWIND AT 4000 FT THAT BLEW US ACROSS THE LOC FOR RWY 26. I IMMEDIATELY REQUESTED THE ILS RWY 27. AT THIS TIME ANOTHER ACFT WAS ALREADY ON THAT APCH AND WE STARTED TO CLOSE IN ON THAT ACFT. THE APCH CTLR TOLD US TO BEGIN A CLB TO 5000 FT AND A TURN; AWAY FROM THE APCH CORRIDOR. AFTER THIS I TOLD MY FO I WAS GOING TO FLY THE ACFT AND EXECUTE THE ILS RWY 26 UTILIZING RAW DATA AND HAVE HIM TO BACK ME UP. THIS APCH AND LNDG WENT UNEVENTFULLY WITHOUT ANY FUTURE PROBS. WE COULD HAVE PREVENTED THIS FROM HAPPENING IF WE HAD ASKED THE CTLR WHICH APCH SHE WAS GOING TO GIVE US. OR; WE COULD ASK FOR A LONGER DOWNWIND; THAT WAY GIVING US MORE TIME TO BRIEF AND PREPARE FOR THE APCH. I COULD HAVE REQUESTED THE ILS RWY 27 LONG BEFORE; DURING THE INITIAL CONTACT WITH THE CTLR.