CARJ EXPERIENCING EQUIP PROBS AND FAILS TO MAKE DSCNT ALT XING RESTR WITH ZID ARTCC.

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ON FERRY AT CRUISE FL350 BTWN ATL AND VXV; THE FMS DUMPED. THE FO WAS THE PF AND WE BOTH WENT GREEN DATA AND RESUMED NORMAL NAV PROCS. I IMMEDIATELY NOTIFIED CTR THAT WE WERE NOW A '/A' ACFT FOR XPONDER PURPOSES. AFTER PASSING VXV; ZID CLRED US TO DSND TO FL280 AND CROSS DRESR INTXN AT FL240. THE FO DID THE MENTAL DSCNT CALCULATIONS AND WE STARTED DOWN. WHILE IN DSCNT; THE FMS CAME BACK. I WENT TO WHITE DATA AND AUTO-TUNED. THE FMS ELECTRONIC DSCNT/VNAV DATA CONFIRMED THAT WE WOULD CROSS DRESR AT FL240. THE FO THEN WENT TO WHITE DATA AND AUTO-TUNED SHORTLY AFTER THAT. THE FMS DISPLAYED 'DR' AND WOULD NOT DO AN AUTOMATIC POS UPDATE. MANUAL POS UPDATES WERE ALSO UNSUCCESSFUL. I AGAIN TOLD ATC WE WERE AN '/A' ACFT AND WE BOTH WENT TO GREEN DATA AND RESUMED NORMAL NAV PROCS. I THEN TUNED THE 'FLM' VOR 117.0 AND SET MY COURSE FOR THE 124 DEG RADIAL IN ANTICIPATION OF FIXING THE TURN AT SWEEP INTXN. EVEN THOUGH THE 'FCC' WAS SELECTED TO THE FO'S SIDE; THE AUTOPLT BEGAN AN UNCOMMANDED L TURN TOWARDS 'FLM.' WHILE DIAGNOSING THE UNCOMMANDED TURN; ZID ASKED US TO INCREASE OUR RATE OF DSCNT. WE HAD BECOME FIXATED ON THE NAV PROB AND THEN REALIZED WE PROBABLY WOULD NOT MAKE THE DRESR XING RESTR. SINCE WE WERE A FERRY FLT WITH NO PAX; WE DUMPED THE NOSE AND HURRIED DOWN TO FL240. WE RECEIVED NO FURTHER QUERIES FROM ZID REGARDING THIS. WE COMPLETED THE FLT IN GREEN DATA WITHOUT INCIDENT. AFTER LNDG; CVG GND CTL ASKED US TO CALL INDY. I CALLED AND EXPLAINED WHAT HAPPENED. HE SUGGESTED IN THE FUTURE THAT I GIVE MORE INFO; IE; EXPLICIT DETAILS ABOUT THE EXTENT OF OUR PROB OTHER THAN SIMPLY RPTING THE CHANGE IN OR XPONDER STATUS. I CONCURRED. HE SAID NO CERTIFICATE ACTION WOULD BE TAKEN.