ACR X TCASII TA RA SAME ALT ASSIGNED; HAD LTSS FROM ACR Y. SYS ERROR.

Narrative

ACR X CRUISING AT FL370; PROCEEDING DIRECT TO ALB. AT APPROX 200 NM W OF ALB; THE CTLR TOLD US TO TURN L TO A 020 DEG HDG FOR TFC AND TO EXPECT TO BE RECLRED DIRECT TO ALB. THE L TURN TO 020 DEG REQUIRED ABOUT AN 80 DEG HDG CHANGE. I PROMPTLY INITIATED THE TURN TO THE ASSIGNED HDG; AT THIS POINT THERE WERE NO TFC RETURNS ON OUR TCASII DISPLAY. NEAR THE END OF THE TURN; (PASSING THROUGH ABOUT 040 DEG HDG) WE GOT POP UP TFC ON OUR TCASII AND AN AURAL TA ('TFC; TFC'). BOTH THE CAPT AND I REDUCED OUR MAP SCALE TO THE 10 MI RANGE AND IDENTED OPPOSITE DIRECTION TFC AT FL370. SINCE WE WERE ESTABLISHED IN THE L TURN AND LINED UP WITH THE CONFLICTING TFC; I DISCONNECTED THE AUTOPLT AND INCREASED THE BANK ANGLE TO INSURE THAT THE TFC WOULD PASS OFF TO OUR R SIDE. AT THIS POINT; ATC REALIZED THE CONFLICT AND TOLD US TO DSND TO FL350 FOR TFC; AND TOLD ACR Y TO TURN L FOR TFC. I IMMEDIATELY INITIATED THE DSCNT TO FL350 AND WE SIMULTANEOUSLY RECEIVED AN AURAL RA ON THE TCASII ('DSND; DSND'). SINCE I HAD ALREADY INITIATED THE DSCNT AS PER THE ATC CLRNC; I COMPLIED WITH THE TCASII COMMANDED DSCNT RATE (ABOUT 1500 FPM). PASSING THROUGH FL360; WE RECEIVED A 'CLR OF CONFLICT' MESSAGE ON TCASII. WE NEVER VISUALLY SAW THE OTHER TFC; BUT; BASED ON TCASII; ESTIMATE THAT IT PASSED OFF OUR R SIDE 1-2 MI LATERALLY AND 800-1000 FT ABOVE US. AS WE LEVELED AT FL350; ZOB CLRED US DIRECT TO ALB. THE TCASII WORKED WELL AS IT GAVE US RA'S IN A TIMELY MANNER TO INSURE SEPARATION.