HVY JET ON VECTORS FOR AN ILS APCH LOSES CONTACT WITH APCH. SEPARATION LOSS RESULTS.

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ON MAY/XA/06 I WAS THE CAPT AND PF ON ACFT X. THE FO WAS HANDLING THE RADIOS. OUR DEST WAS MEM; AND WE WERE FLYING THE WLDER 4 ARR. AT CLARK INTXN WE TURNED TO A HDG OF 175 DEGS; COM WAS NORMAL ON 125.8. WE WERE SENT TO THE FINAL CTLR ON 120.92. THIS CTLR GAVE US A DSCNT TO 3000 FT. AT THAT TIME WE WERE APPROX ABEAM MEM. DURING THE DSCNT TO 3000 FT; WE DID NOTICE THAT THE FREQ WAS QUIET. INITIALLY IT DID NOT ALERT US THAT ANYTHING WAS WRONG; SINCE WE HAD JUST CHKED IN ON THE FREQ AND DID NOT KNOW HOW MUCH TFC HE WAS CTLING. AFTER ABOUT 1 1/2 MINS; THE FO SAID 'IT SURE IS QUIET.' I TOLD THE FO TO CHK BACK IN THE CTLR; HE ASKED 'HOW DO YOU HEAR ME?' THE FO REPLIED 'LOUD AND CLR.' THE CTLR STATED THAT HE HAD BEEN TRYING TO GET A HOLD OF US; THEN GAVE US VECTORS FOLLOWED BY A CLRNC FOR THE ILS RWY 36R APCH. NOTHING ELSE WAS SAID! UNKNOWN TO US; THE CTLR WAS TRYING TO WORK ANOTHER ACFT Y ON A BASE ENTRY AHEAD OF US (WE WERE ON DOWNWIND). THIS WOULD HAVE WORKED; EXCEPT HE ALLOWED ACFT Y TO DEVIATE; CREATING THE POTENTIAL CONFLICT. THE CTLR TRIED TO REACH US 3-4 TIMES AT 20 SECOND INTERVALS (DID NOT TRY 121.5; WHICH WE WERE ON); AND DURING THAT TIME DID NOT GIVE ACFT Y ANY DIRECTIVES. AFTER THE ATTEMPTS TO REACH US; HE DID GIVE ACFT Y A HDG; BUT WE AND ACFT Y GOT SLIGHTLY LESS THAN 3 MI FROM EACH OTHER. IN A STERILE COCKPIT; IT IS STILL POSSIBLE TO MISS A RADIO CALL. IT IS CONCEIVABLE TO MISS 2 RADIO CALLS; BUT NOT 3 OR 4 CALLS. AFTER 2 CALL ATTEMPTS IN THE APCH ENVIRONMENT; THE CTLR HAS GOT TO ASSUME LOST COM AND TAKE THE APPROPRIATE ACTION.