F5 loadbalancer and session timeout in ASP application

You're right, http is stateless and tcp is stateful.

You said the issues occur since you've changed from a Microsoft product to a F5, so I would search here for the differences. It's not new to me that Microsoft has it's own interpretation of RFCs.

If the problem is reproducible, I would recommend a Wireshark trace on the app server and client and compare them. Maybe you need to do further investigation on the "middle-ware" (like the F5 or Firewall) if the traces don't corelate.