I ran into this the other day, my ECS instances were dieing off and docker ps showed Exited (137) About a minute ago. Looking at docker inspect I noticed:

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"State": {
 "FinishedAt": "2015-09-20T21:38:58.188768082Z",
    "OOMKilled": true
  },

This tells me that ECS / Docker enforced the memory limit for the container, and the out-of-memory-killer killed off the contained processes. Raising the ECS memory limit for this process resolved the issue.