The issue is quite similar to MDL-59226. However, ajax calls mentioned there are different to what has been found for this issue.

A site where it's been reported has a big number of notifications and almost all of the notifications are related to a small number of admins (automatic notifications on new users, self enrolments etc).

The issue is happening for admin users after a few hours of being offline. First /my/ call takes about 3 minutes to load the page, but all further pages including logout/login attempts - all work fine. Then, after a few hours of not working on the site the issue is re-ocurring.

We could profile this and core_message\api::count_unread_conversations turned out to be the cause. See profiling graph attached.

Having this SQL we could replicate the issue directly in db (MySQL 5.7). First query takes ~3 minutes, but then results seem to be cached and the query completes almost instantly.