Heroku web dynos timing out inside BufferedLogger#write

Heroku web dynos timing out inside BufferedLogger#write

Last night one of our web dynos experienced problems and stopped
responding. It is a unicorn process. One by one the four child processes
exceeded the Rack:Timeout limit, in each case the process was inside a
call to BufferedLogger#write. From then on, nearly all requests to the
dyno timed out with a H12 error, until the dyno was manually restarted 4
hours later.
We have experienced this issue a number of times now.
The fact that the processes all timeout while in the same method 'write'
inside a call to 'flush' tells me that something is blocking the write.
However the exception is written to the log.