FLC OF AN MLG ACR ACFT TOOK EVASIVE ACTION IN RESPONSE TO A TCASII RA DUE TO CTLR WORKLOAD DISTR.

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CRUISING AT FL310; DEVIATING AROUND NUMEROUS TSTMS. ZKC SWITCHED US OVER TO ZDV. CHKED IN WITH ZDV; GAVE OUR ALT AND NOTIFIED THEM WE WERE DEVIATING FOR WX (PREVIOUSLY APPROVED BY ZKC). DID NOT RECEIVE A RESPONSE FROM ZDV; ALTHOUGH WE COULD HEAR THE CTLR TALKING TO OTHER ACFT. SEVERAL OTHER ACFT ALSO ATTEMPTED TO CHK IN WITH NO RESPONSE FROM CTR. THE FREQ WAS VERY CONGESTED DUE TO NUMEROUS ACFT REQUIRING WX DEVS. WE MADE NUMEROUS ATTEMPTS TO CHK IN; INCLUDING IDENTING ON OUR XPONDER. WE HAD OUR TCASII ON; AND NOTICED AN ACFT ALSO AT FL310 ABOUT 10-12 MI AHEAD OF US GOING FROM OUR R TO L. WE TURNED APPROX 20 DEGS R TO INCREASE SEPARATION AND PASS WELL BEHIND HIM. ABOUT THE SAME TIME THE CTLR FINALLY CONTACTED US AND GAVE US INSTRUCTIONS TO TURN R ALSO; WHICH WE WERE ALREADY DOING. OUR CLOSEST POINT WITH THE OTHER ACFT WAS ABOUT 6-8 MI. WE DID NOT GET ANY TCASII TA'S; AS WE WERE NOT THAT CLOSE. THE CAUSE OF THIS INCIDENT CAN BE ATTRIBUTED TO THE HVY WORKLOAD THE CTLR HAD ALONG WITH TOO MANY XMISSIONS ON THE FREQ; CAUSING MANY OF THEM TO BE BLOCKED OUT. THE TCASII PROVED INVALUABLE. WITHOUT IT; OUR CLOSURE DISTANCE WITH THE OTHER ACFT WOULD HAVE BEEN MUCH CLOSER.