We are using SQL 2K with sp4. Our replication is using Push method. Both
publisher and distribution database are on the publisher server.

Once in a while, our distribution agent is getting a timeout and the message
is "The process is running and is waiting for a response from one of the
backend connection". The timeout happened after 10 minutes of waiting for
the backend connection.

What does this message mean? Is it a networking issue or the subscriber
server is busy of doing something? If you can, please lay out all the
possible scenerio to make this error happen.

Will increasing the query time on distribution agent profile solve this
problem or just prolong the problem? Any help on this is very much
appreciated.

The easiest thing to do is to run sp_who2 and dbcc inputbuffer to see
exactly what the agent is doing. Keep an eye out for blocking.
For an initialization I'd increase the querytimeout as a matter of course on
large databases - I often see that simply creating the indexes will take 30
mins or so.
Cheers,
Paul Ibison SQL Server MVP, www.replicationanswers.com .