Ref T11458. Currently, we're very aggressive about closing connections.

I want to change things so we only close connections we haven't used recently (for example: no activity in the last 15 seconds). This should let us do a better job of holding open approximately only the connections we actually need in a wider range of cases.

Ran some code that uses queries.

This doesn't do anything on its own; see next diff in Phabricator for meaningful changes.