Author : if you have problems with this, better ask me than the author of OverpassAPI (Roland)
More information here :

If runing under debian you can use that SysV start-up script.

Change the variables at the begining to match your installation (DB_DIR,
You can also set those variables in the /etc/default/overpass file

This script runs the dispatcher at start up and the diff update process.
You first need to update or create in your $DB_DIR the replicate_id file with the
replicate sequence number corresponding to the date of the OSM file you
imported if your are running it for the first time

INSTALL for squeeze/wheezy

cp overpass /etc/init.d/
update-rc.d overpass defaults