Cookie persistence should be the primary with source address as a fallback. Hopefully that's how it's setup.
When you leave a session idle for 10 minutes, the TCP connections involved are closed long before that, by your browser if not by the server. Session persistence is preserved through cookies assigned by SAP. It sounds to me like the cookie is not being properly handled , so I would check your session persistence setup under the Virtual Server in the BIG IP.
A way you can verify this is to use a web browser plugin, such as Firebug or HTTPWatch (paid version) to look at your cookies going through the BIG-IP and going direct to the server, and compare the cookies to see if there are any differences.
Without knowing more about your configuration, I would focus on the cookies as the source of the issue.