ACR X TCASII RA WITH Y NON ADHERENCE TO ATC CLRNC HAD LTSS FROM Y. SYS ERROR.

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I WAS THE LOW ALT CTLR WORKING ACR X INBOUND TO MIA FROM ENGLAND. ACR Y CLBING OUT FROM MIA TO CURACAO. UPON REALIZING THE CONFLICT; I ISSUED DSCNT TO FL 200 (AMENDED FROM DSCNT TO 16000 FT) TO X -- NO RESPONSE. ISSUED Y CLB TO FL 190 (AMENDED FROM CLB TO FL230). ISSUED DSCNT TO FL200 AND HDG 180 DEGS (70 DEGS L) TO X. READS BACK þHDG 190 DEGS AND DO WE HAVE TFC BELOW US?' I ACKNOWLEDGED AND ADVISED MAINTAIN FL200 AND FLY HDG 180 DEGS. X'S MODE C INDICATES STILL DSNDING THROUGH FL200; SO I TURN ACR Y 35 DEGS L (HDG 090 DEGS). Y ADVISES 'TFC IN SIGHT.' AFTER LATERAL SEPARATION IS ESTABLISHED; X IS AT FL180 DSNDING TO 16000 FT. I ASKED IF HE HEARD ME CLR HIM TO FL 200? NEGATIVE. AFTER BEING RELIEVED DUE TO LOSS OF SEPARATION (3.25 NM AND 100 FT) I FELT THAT X WAS UTILIZING TCASII INFO BECAUSE OF HIS VERBAL AND X'S PERFORMANCE RESPONSE. HE NEVER ADVISED ME OF RESPONDING TO A TCASII RA. WE TALKED TO THE PLT AFTER LNDG AND HE ADVISED THAT HE WAS INDEED RESPONDING TO A DSND RA OF 2200 FPM OR GREATER. I SUPPORT TCASII AS A COLLISION AVOIDANCE TOOL; BUT BY NOT ADVISING OF RA MANEUVERING; CTLRS WILL UTILIZE EXTREME CTL ACTIONS TO TRY AND PREVENT COLLISION. THIS MAY LEAD TO 'LOSING CTL' OF OTHER PORTIONS OF THEIR SECTOR. I ALSO FEEL FOR THE PLT. DURING AN RA; HE/THEY ARE TRYING TO ACQUIRE THE TCASII TFC; RESPOND TO AND PERFORM THE RA MANEUVER; FLY THE ACFT; COM WITH ATC. THE WORKLOAD IS SIGNIFICANT. I THINK; IN THIS CASE; HE 'FORGOT' TO ADVISE ME OF THE RA AND DIDN'T 'HEAR' FL200 (3 TIMES) BECAUSE OF THE WORKLOAD AND FATIGUE (ENGLAND TO MIA). SUPPLEMENTAL INFO FROM ACN 279191: X; WHILE DSNDING TO 16000 FT GOING DIRECT TO JUNIOR AT 8000 FT; WE RECEIVED TCASII RA ALERT. TCASII DIRECTED US TO DSND AT 2000 FPM. WE WERE ALREADY DSNDING AT 2200 FPM FROM MIA APCH. MIA APCH THEN CLRED US TO TURN L TO A 180 DEG HDG FOR TFC. WE TURNED TO A 180 HDG AND THE TCASII RPTED WE WERE CLR OF TFC. WE CONTINUED DSCNT AND RPTED WE WERE CLR OF TFC. WE CONTINUED DSCNT TO 16000 FT. MIA APCH THEN ASKED IF WE HEARD A CLRNC TO LEVEL AT FL200. WE ADVISED MIA APCH. HE ADVISED ME THEY CHKED THE APCH CTL RECORDING TAPE AND IT SHOWED WE NEVER ACKNOWLEDGED A CLRNC TO LEVEL AT FL200. SUPPLEMENTAL INFO FROM ACN 279190: X; WHILE DSNDING INTO MIA CLRED TO 16000 FT WE RECEIVED A TCASII TA FOLLOWED BY AN RA TO MONITOR VERT SPD NOT LESS THAN 2000 FPM. WE WERE ALREADY DSNDING AT 2200 FPM. THE CTLR GAVE US A L TURN TO 180 DEGS AND ASKED WHY WE DID NOT LEVEL AT FL200 AS REQUESTED. WE DID NOT RECEIVE OR ACKNOWLEDGE THIS CLRNC. WE ALSO DID NOT SEE THE OTHER ACFT. ON THE TCASII SCREEN -- ON 40 NM RANGE; THE TARGET LOOKED TO BE WITHIN 1 MI. THE FO WAS NOT IN HIS SEAT AT THE TIME. I WAS THE IO ON THIS TRIP.