C560 WITH ZMA AT FL240 DEVIATES AROUND WX; WITHOUT CTLR APPROVAL BECAUSE OF FREQ CONGESTION AND NON RESPONSE AND IS QUESTIONED BY ATC.

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WHILE ON AIRWAY 699 TO PERMT INTXN AT FL230 WE SAW A LARGE BUILD-UP AHEAD. I CALLED ZMA 133.40 REQUESTING A DEV TO AVOID THE BUILD-UP (WE HAD GONE THROUGH A SMALLER ONE SHORTLY BEFORE THAT HAD MODERATE TURB). ZMA CAME BACK BUT DID NOT ACKNOWLEDGE OUR REQUEST BUT HANDED US OFF TO NEXT CTLR ON 135.07 (ALSO ZMA). ATC IN THIS AREA WAS EXTREMELY BUSY BUT THERE WERE ONLY A FEW BUILD-UPS HERE AND THERE; OURS JUST HAPPENED TO BE ON THE AIRWAY. THE NEXT FREQ WAS JUST AS BUSY AS THE FIRST. AFTER CHKING IN WITH NEXT CTLR I TRIED 3 TIMES TO REQUEST A DEV FOR WX; NO ACKNOWLEDGEMENT. WE WERE THEN JUST AT THE POINT WHERE WE WERE ABOUT TO GO INTO THE WX AND WERE GIVEN A CLB TO FL290. WE STARTED A VERY SLOW CLB AS THE WX WAS JUST ABOVE US. I AGAIN REQUESTED DEV AND GOT NO RESPONSE. (I DON'T BELIEVE I WAS BEING IGNORED; I JUST THINK I MUST HAVE BEEN STEPPING ON THE CTLR AND OTHERS ON THIS VERY CONGESTED FREQ.) WE REACHED PERMT INTXN AND INSTEAD OF TURNING L ONTO AIRWAY MY CAPT CONTINUED THE L TURN TO 320 DEGS TO AVOID THE WX. WE SHOULD HAVE BEEN HDG ABOUT 017 DEGS. AFTER APPROX 1 MIN ATC ASKED OUR HDG. I TOLD HIM 320 DEGS DUE TO WX. HE WAS UPSET. I TOLD HIM WE HAD TRIED NUMEROUS TIMES TO REQUEST A DEV BUT WERE UNABLE TO GET A RESPONSE. HE SAID THAT WAS NO REASON TO DEVIATE WITHOUT PERMISSION DUE TO THE POTENTIAL TFC CONFLICT. ABOUT THIS TIME WE WERE PAST THE WX; HE TOLD US TO TURN R TO 060 DEGS AND LEVEL OFF AT FL240 (AS WE WERE CLOSE TO FL240; WE WENT UP TO FL242 FT FOR A MOMENT). WE WERE SOON BACK ON AR7 TO DIXON WITH NO FURTHER PROBS. CONTRIBUTING FACTOR (TO ME THIS WAS THE MAIN FACTOR): AN INORDINATE AMOUNT OF CONGESTION ON THE FREQ. FACTORS AFFECTING THE QUALITY OF HUMAN PERFORMANCE: ONE OF OUR PAX WAS EXCEPTIONALLY CONCERNED ABOUT THE SLIGHTEST BIT OF TURB. ACTIONS OF INACTIONS: I WAS NOT COMFORTABLE WITH SUCH A LARGE TURN TO THE L WITHOUT PERMISSION AND DID VOICE THIS CONCERN BUT I FEEL I WAS NOT ASSERTIVE ENOUGH.