B737 FLT CREW IN ZOB AIRSPACE CONTACTS ATC SEVERAL TIMES WITH NO RESPONSE AND ENDS UP RESPONDING TO A TCAS RA CAUSED BY CONVERGING TFC.

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WE WERE LEVEL AT FL360; AND HAD BEEN FOR A WHILE. ABOUT 120 MI E OF THE CRL VOR ON J554; WE RECEIVED A FREQ CHANGE TO ZOB ON 120.07. I SWITCHED OVER; AND NOTICED THE FREQ WAS BUSY. THE WX WAS CLR WHERE WE WERE; BUT APPARENTLY THERE WAS A LOT OF DEVIATING TFC NEAR CLE. MY FIRST CHK-IN WENT UNANSWERED; SO I FIGURED THE CTLR WAS BUSY. I WAITED 30 SECONDS OR SO AND TRIED AGAIN; BUT GOT NO ANSWER. I WAITED 30 OR 40 SECONDS TO TRY AGAIN. JUST AFTER MY THIRD CHK-IN ATTEMPT; WHICH WENT UNANSWERED; WE RECEIVED A TA ON OPPOSITE DIRECTION TFC ABOUT 12-15 MI AHEAD; +500 FT AND DSNDING. AT ABOUT 9 MI; WE RECEIVED A CLB RA OF AT LEAST 1500 FPM. AT THAT TIME; THE TFC WAS ABOUT +200 AND STILL DSNDING. WE DISCONNECTED THE AUTOPLT AND COMPLIED WITH THE RA. I CALLED ATC AND RPTED WE WERE IN A TCAS CLB; WHICH STILL WENT UNANSWERED. IT HAPPENED VERY QUICKLY; AND I ESTIMATE THAT AT THE 5 MI POINT; THERE WAS ABOUT 1000 FT OF SEPARATION. WE THEN RECEIVED A 'MONITOR VERT SPD;' AND THEN A 'CLR OF CONFLICT.' WE WERE ABOUT FL366. I CALLED ATC AND RPTED THAT WE WERE DSNDING BACK DOWN TO FL360; AND HE REPLIED ROGER. WE NEVER HEARD ANOTHER THING ABOUT IT.