BEECHJET ACFT ON STAR WAS INSTRUCTED TO KEEP SPD UP BY CTLRS AND CROSS INTXN AT 10000 FT. APCH CTLR CLRED FLC TO DSND; BUT THE SPD WAS STILL 300 KTS SO RPTR CAPT SLOWED PRIOR TO DSCNT. APCH CTLR TOLD FLC THAT THEY SHOULD HAVE SLOWED IN ACCORDANCE WITH THE STAR RESTR.

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ON OUR INITIAL DSCNT OUT OF FL430; THE CTLR ASKED US TO MAINTAIN OUR BEST FORWARD SPD FOR SPACING. WE WERE TOLD TO ADVISE NEXT CTLR OF OUR ATC AIRSPD ASSIGNMENT. THAT CTLR ALSO CONFIRMED BEST FORWARD AIRSPD. WE WERE THEN PASSED ON TO 1 OR 2 OTHER CTLRS DURING OUR DSCNT INTO HOU ON THE ROKIT 1 ARR. WE WERE GIVEN 'CROSS SSLAM AT 10000 FT.' AS WE PASSED OVER DUUNK WE WERE STILL MAINTAINING BEST FORWARD AIRSPD SINCE THAT WAS THE LAST ATC ASSIGNED AIRSPD REQUEST. AT SOME POINT HOU APCH CLRED US DOWN TO 7000 FT. I WAS IN THE PROCESS OF SLOWING THE ACFT FROM 300 KTS TO 250 KTS BEFORE STARTING DOWN TO 7000 FT. THE CTLR THEN ASKED WHAT OUR AIRSPD WAS. WE INFORMED HIM WE WERE SLOWING FROM 300 KTS TO 250 KTS BEFORE DSNDING. HE QUICKLY ADVISED US THAT THE ARR SHOWED A PUBLISHED 250 KTS OVER DUUNK. WE TOLD HIM WE HAD PREVIOUSLY BEEN ASSIGNED 'BEST FORWARD AIRSPD' AND HAD NOT BEEN TOLD TO 'RESUME NORMAL SPD' OR GIVEN A XING AIRSPD RESTR. THE CTLR THEN MENTIONED SOMETHING THAT THE PREVIOUS CTLR HAD FAILED TO GIVE US NORMAL SPD BUT INSISTED THE ARR CALLED FOR 250 KTS REGARDLESS. WE NEVER QUESTIONED THE CTLRS ABOUT OUR AIRSPD DURING OUR DSCNT ON THE ARR AND NONE OF THEM EVER ASSIGNED US ANYTHING OTHER THAN 'BEST FORWARD SPD.' TO AVOID A SIMILAR SIT IN THE FUTURE; I WILL TRY TO VERIFY WITH EACH SUBSEQUENT CTLR IF WE ARE TO CONTINUE WITH 'PREVIOUSLY ASSIGNED.'