Why is "Checking whether to disconnect long-running connections for process xxxxx" showing in the apache error log?
This is actually not an error but a normal occurrence when using mod_passenger. This message check to see which application processes have been idle long enough before being shut down to conserve memory.
If you wish to increase this idle time, you can do so by adding "PassengerPoolIdleTime XXX" where XXX is the number of seconds before the process hits this limit. This can be added via include file which the following article shows how you can add them:
How to Edit the Apache Configuration File