When this error happens, if you try to start the JBoss AS service again, is it successful? Or do you have to first do something with PostgreSQL?

If PostgreSQL is taking too much time to start up properly, then the start of the JBoss AS service needs to be delayed until PostgreSQL is fully functional. Note that some services tell Windows that they are running before they are fully functional. Apparently, PostgreSQL does this (most of the tools used to run Java apps as services do the same thing - JBoss AS is rarely up and running when Windows thinks it is).

Are you using PostgreSQL as your DefaultDS? If not, can you change the configuration for your apps to not establish the database connections until the first user request? Or simply add in a delay before getting the connections (if you have a startup mbean doing this).

What tool are you using to run JBoss AS as a service? Perhaps that tool provides a mechanism to delay the startup.