Date: 201210
Local time of day: 1201-1800
Location: ZZZ.ARTCC, US
Flight conditions: VMC
Light: Daylight
Aircraft: B737-300
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Cruise
Airspace: Class A ZZZ
ATC / advisory: Center ZZZ
Aircraft component: Fuel System
Component problem: Malfunctioning
Reporter organization: Air Carrier
Reporter function: Captain; Pilot Not Flying
Human factors: Troubleshooting; Workload; Training / Qualification
Second reporter function: First Officer; Pilot Flying
Anomaly: Aircraft Equipment Problem Critical; Inflight Event / Encounter Fuel Issue
Detector: Person Flight Crew
When detected: In-flight
Result: Aircraft Equipment Problem Dissipated; Flight Crew Diverted; Flight Crew FLC complied w / Automation / Advisory; Flight Crew Inflight Shutdown; Flight Crew Landed As Precaution; Flight Crew Overcame Equipment Problem; Flight Crew Took Evasive Action; General Maintenance Action
Contributing factors: Aircraft; Procedure
Primary problem: Ambiguous

Narrative: We were enroute at FL350. I decided to burn the last bit of remaining fuel (approximately 300 LBS) in the center tank. I announced this to the First Officer and said I would use the crossfeed procedure (open the crossfeed and turn on one center pump). Several minutes went by and the First Officer said the center tank was empty. I noticed a slight imbalance of about 300 pounds so I turned the center tank pump switch off and left the crossfeed open. Then; I turned off the low side pump (#1 engine) to balance and announced this as well to the First Officer. He acknowledged. Several more minutes went by when I noticed the #1 engine instruments start to roll back. As the airplane started to roll left; the First Officer got on the controls and corrected the flight path. I called Center and told the Controller our engine had rolled back and we needed an immediate descent. We were given clearance to descend (we were descending anyway). While descending; I turned the #1 engine fuel pumps back on and closed the crossfeed valve. All bright/dim sequences seemed to be working properly. I told the First Officer to take the COMM 1 radio and keep flying the plane while I looked for a place to land. On the Fix page; we were only 50 miles from a major airport so I told the First Officer that's where we were headed. The 'A' Flight Attendant called to say the cabin lights went out. I explained that our left engine had rolled back and as soon as I could start the APU; the lights should come back. We were still descending and low enough to start the APU; so I did. While the First Officer was coordinating our divert with ATC; I got on the Passenger Address System and told our customers what had happened. According to the flight attendants; it was in no way alarming; but reassuring. I explained the need to land as soon as we could; but that once we did land; we would simply taxi back to the gate and go from there. Still descending; we decided to run the Engine Failure/Shutdown Checklist as throttle movement was unresponsive on the #1 engine. When the engine was shut down; I noticed all engine instruments were indicating the same as when it had rolled back: not frozen; but windmilling. I called the divert airport Operations to let them know we were coming in and for them to please call Dispatch. We were being vectored by Approach and wanted to see if we could restart the #1 engine because it met all the parameters. We ran the In-flight Engine Start and got our engine back. We also ran the Descent Checklist and told Approach we were ready. We got cleared for the visual and landed and taxied without incident. I made an entry in the Maintenance log and called Dispatch. We just happened to have one of our Mechanics on board. He stated that Boeing might come out with something on high altitude cross-feeding or simply not crossfeed unless there was a serious imbalance. Speaking to my Chief Pilots after the incident; they also recommended not crossfeeding at high altitude. I don't normally crossfeed with such a small imbalance and; after this incident; I never will. Can we recommend the same advice to other Crews? There is nothing in our books that tells us not to; but look what could happen.


Second narrative: [Narrative #2 contained no additional information.]


Synopsis: A B737-300 engine shutdown from fuel starvation at FL350 while the engine was being cross fed with the high tank pumps ON and the crossfeed valve open. The flight diverted to a nearby airport and the engine was restarted prior to landing.