A very interesting thread! I would also like to hear of any stats on engines that successfully relit having just 'failed'. I suspect not many but am open to correction. I agree with the view that modern engines don't 'fail' for any reason.
I'm perplexed by your post. How on earth are you defining a
failure .
Of course engines malfunction for a variety of reasons and some of these malfunctions require the pilot to secure the engine
in other words shut off the fuel In spite of this the pilot always has the discretion to restart an engine which he has
secured although he would rarely do this unless he has had other engines losing power as well.
All told the multiple engines out that had the time to be restarted have been very few. A couple of accidental fuel shutoff events, a couple of mistaken shutdowns at altitude and a couple of volcanic ash events. Perhaps we might also include the Kegworth event.
There have been numerous events with multiple engines out but no attempt or time was available to restart.
As long ast the high pressure compressor will spin and still has a turbine behind it, most engine will restart and run at idle if fuel is still available..
Running at high power is another thing however.