“using stale statistics instead of current ones because stats collector is not responding”

To temporarily alleviate the problem until you find a solution, have you tried sending a SIGHUP to the server when it occurs?

I have a similar problem in a fairly different situation: the database runs on a windows desktop PC together with Apache and proprietary machine control software. My case fits in the "slow or overloaded" category. However, I see this happening even at night while the machine is idle. I've been monitoring many system parameters with PRTG to try to find a link with a specific activity, but I can only see the consequences, not the cause.

I don't see this as a simple warning because, once it starts, queries that normally take 100ms will start taking 10 seconds (but still leave other queries unaffected). Also, it stays that way until I send a SIGHUP (pg_ctl.exe reload ...), which I think is weird since the process should resume once it is no longer starved.

P.S. please don't hail me about the windows desktop part: I know, and ... circumstances.

This basically means your (virtual?) machine is too slow or overloaded, because the statistics collector background task is being starved. It is running with a lower priority by default, because you do not want it to disrupt normal database operations (too much).

If you can afford it, add some more CPUs to it and make sure you have plenty of RAM.

Technically though, this is not a severe error - your queries will run, but may be using less than optimal execution plans because they did not see an update to table statistics in a while.

If you want to, you can always force an update of statistics for the most commonly changed tables by running the ANALYZE or VACUUM ANALYZE statements on those.