OK, so, let's put this into perspective. You'd like complete end-to-end visibility of every packet or connection (or lack thereof) between you and your client, at every hop under your control, between your server and the 'client'. Forgive my negativity but people have built careers and companies around such things.
On a more positive note, what can you do?
- You could perform a tcpdump on the F5 over a suitable period of time, dumping it to a file and then use Wireshark or a similar commercial application to perform some analysis (manual or otherwise).
- Check relevant switch ports and network interfaces for errors and drops.
- Take a close look at the Traffic Summary statistics (of course, the downside is these stats are global, not VS specific)
- Take a close look at the Profiles Summary > TCP statistics (of course, the downside is these stats are global, not VS specific)
- Take a close look at the Network statistics (of course, the downside is these stats are global, not VS specific)
- Enable TCP reset logging and monitor statistics accordingly
There are also other things you could consider that may help you track things down (or simply improve things in general);
- Can you monitor server connection/CPU/memory usage?
- Consider modifying TCP profile settings based on the traffic type
- Consider increasing default idle-timeout settings
- You could perhaps use an iRule to log relevant VS specific statistics?
- You could use increased SNMP monitoring and monitor accordingly
- You could setup sFlow and monitor accordingly
If you can provide some more info on the VS type and setup, traffic flow and behaviour (constant, small flows, large bursts etc.) I may be able to offer further advice.