A BE9L PLT; AFTER MAKING A MISSED APCH AT SVC; EXPERIENCED A R ENG FLAMEOUT DUE TO FUEL STARVATION.

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I EXPERIENCED A R ENG FLAMEOUT WHILE ENRTE TO DEMING; NM. THIS OCCURRED AFTER A MISSED APCH DURING LOW IFR CONDITIONS AT SILVER CITY; NM. AFTER FOLLOWING THE CHKLIST PROCS FOR A FAILED ENG; I FLEW THE REMAINING DISTANCE TO DEMING (DMN) AND LANDED SAFELY. AFTER A FLT TO SCOTTSDALE; AZ; I WAS OF THE OPINION THAT THERE WAS SUFFICIENT FUEL ON BOARD TO MAKE THE 1 HR RETURN FLT TO SILVER CITY AND LEAVE THE AIRPLANE WITH A NORMAL SUMMER CAPACITY OF APPROX 1400 LBS OF FUEL. CONTRIBUTING FACTORS TO THE ENG FLAMEOUT INCLUDE A NEARLY 20 MIN GND WAIT IN SCOTTSDALE FOR IFR RELEASE. ANOTHER FACTOR WAS AN UNANTICIPATED DEP RTE THAT ADDED NEARLY 25 MORE MINS TO THE FLT-PLANNED RTE. THE WX FORECAST WAS MARGINAL VFR CONDITIONS FOR THE TIME OF ARR IN SILVER CITY. ONCE ENRTE; I CONTACTED FLT WATCH FOR AN UPDATE ON THE WX. I WAS INFORMED THAT THE LATEST AUTOMATED OBSERVATION INCLUDED A TSTM IN THE AREA WITH A CEILING OF ABOUT 6000 FT OVCST AND LESS THAN 2 MI VISIBILITY. ONCE THE FLT WAS 'IN RANGE' OF SILVER CITY; I AGAIN RECEIVED A WX UPDATE; THIS TIME FROM THE LCL AWOS. THE CONDITIONS WERE SIMILAR TO THE PREVIOUS BRIEFING; WITH THE EXCEPTION OF A REDUCED VISIBILITY OF 1-3/4 MI. THE VOR-A REQUIRES ONLY 1 MI VISIBILITY; SO I DECIDED TO FLY THE APCH. UPON REACHING MDA; I REALIZED THAT THERE WAS VERY LITTLE CHANCE OF SEEING THE RWY ENVIRONMENT; BUT CONTINUED THE APCH TO THE MISSED APCH POINT. AFTER FLYING THE PUBLISHED MISSED AND RETURNING TO VISUAL CONDITIONS S OF SILVER CITY; I ADVISED ZAB THAT I WOULD BE PROCEEDING VISUALLY TO DEMING ARPT. CONSIDERING THE IMPROVING WX CONDITIONS BTWN SILVER CITY AND DEMING; I CANCELED IFR APPROX 25 MI FROM DEMING AND BEGAN MONITORING CTAF. THE FLAMEOUT OCCURRED A FEW MINS LATER. IN HINDSIGHT; I REALIZE THAT THE FINAL CONTRIBUTING FACTOR WAS THE UNRELIABILITY OF THE FUEL QUANTITY INDICATORS. AT THE TIME OF THE FLAMEOUT; I STRONGLY BELIEVED THERE TO BE AT LEAST 1200 LBS OF FUEL. A LATER FULL REFUELING SHOWED THAT THERE WAS APPROX 40 GALS REMAINING IN THE L FUEL SYS. I CLRLY UNDERSTAND THAT THE RESPONSIBILITY OF THE FLT IS MINE. IT IS NOW MY RESPONSIBILITY TO MAKE SURE THAT THIS IS AN ISOLATED INCIDENT.