If I use the same browser with the same machine there is no issue. Turning it on and off, closing the browser etc all seems to be fine.
If I use different browsers on the same machine, it logs me out between browser sessions. (I accept this is a weird thing to do, but is probably what highlighted the issue for me. Just seeing if Chrome is any good, really.)
If I use different machines, I get logged out, as you described. I would suggest this is something that needs looking into. I regularly use three different machines during the course of a normal working day.