Of HTTP-over-SSL/TLS connection persistence

Ok, so now I'm postulating a server-side gizmo speaking "HTTP 1.0 | 
1.1"-over-SSL/TLS that tries its best to have such connections be persistent.

What do browsers typically do in ~this~ case -- how long before they'll 
timeout an idle connection if the user causes it to connect and then walks 
away for whatever reason?

thanks,

JeffH