LOSS OF ALL ACFT THRUST AND TCASII WARNING ON ANOTHER ACFT.

Narrative

SE OF DALLAS ENRTE TO ALEXANDRIA; WE APCHED A CLOUD DECK AT FL370. I ASKED THE COPLT TO TURN ON THE ENG INLET HEAT PRIOR TO ENTERING THE CLOUDS. HE RESPONDED THAT THERE WASN'T MUCH OUT THERE YET; AS WE COULD SEE THE GND. A FEW MOMENTS LATER I REQUESTED ENG INLET HEAT AND IT WAS TURNED ON. AS WE ENTERED THE CLOUDS I BEGAN SPD REDUCTION DUE TO TURB. WE REQUESTED DSCNT TO FL330 IN HOPES FOR SMOOTHER AIR. I USED SPOILERS DURING THE DSCNT TO STABILIZE AIR SPD. I REQUESTED A LOWER ALT AND COULD NOT DSND AT THAT TIME BECAUSE OF TFC. UPON REACHING FL330; I STOWED THE SPOILER AND ADVANCED PWR. I HAD AN INCREASE IN ITT AND NO INCREASE IN N1. THE ACFT CONTINUED DECREASING ON AIR SPD. WE DECLARED AN EMER AS AIR SPD CONTINUED TO DECREASE. THE L ENG CONTINUED TO SPOOL DOWN AND FLAMED OUT AT APPROX FL310. AT THIS TIME THE R ENG STARTED TO SPOOL DOWN AND SHORTLY THEREAFTER FLAMED OUT. ADG DISPLAYED AND WE CONTINUED DSNDING. THE APU WAS STARTED AT ABOUT FL200 AND THE R ENG STARTED AT APPROX 15000 FT. THE L ENG STARTED AT 9000 FT AFTER SEVERAL ATTEMPTS. WITH PWR RESTORED AND VECTORS FROM ATC; WE LANDED AT SHREVEPORT. SUPPLEMENTAL INFO FROM ACN 286427: DURING CRUISE; FL310 CLOSE TCASII WARNING AT 2700 FT ABOVE; RAPID DSCNT THROUGH OUR ALT -- CAUSED MOMENTARY WARNING TO 'MONITOR VERT SPD' -- HAPPENED SO FAST TCASII COULD NOT COMPUTE AN ALT CHANGE COMMAND. HAPPENED WITHIN SECONDS -- INITIALLY; IT WAS THOUGHT TO BE A FALSE WARNING. APPARENTLY; A CORPORATE JET AT FL370 LOST BOTH ENGS AND EXECUTED A RAPID DSCNT.