What could cause an apache server to lock up with "sending reply" processes?

Solution 1:

Looks like those connections get stuck for a long while (SS is the time the request has taken, and some of those are pushing several minutes).

My instinct tells me to look at the database and the PHP application. Check so that you have enough available connections in the pool, check for maintenance tasks (full vaccum can lock the database for a long period of time!) and log long queries to see if you are doing something that may lock an important table. It can also be a problem in the PHP script that prevents it from terminating in a timely manner.

Here is a page with some useful debugging tips for this kind of situations.