I have two HP Computers running Windows XP home version and both suffer from the same problem that I have never noticed when using any other computer brand. After a period of time connected to the Internet, each system apparently loses its ability to recognize that the system is still connected to the Internet via a dial-up connection and all Internet activity ceases; that includes e-mail sending and retrieval, and use of any browser. The browsers, whether it be Explorer, Firefox, Netscape or Opera are unable to connect to any URL. The only solution that I have found is to completely reboot the system. Logging off and then back on again and re-dialing does not resolve the problem.

Does anyone have a suggestion as to a fix?

I would suspect your modem is doing that.

how about simple timing out?

is your services on?  One time I accidently shut it off and then I turned it back on then if fixed the problem.

I would go the same way as suggested by previous posters: check via Control Panel settings of Network Connections -> properties, and Hardware -> modem properties.
Dialup modems do, as a rule, hang up "after some time" (how long is that in your case?), but you can change that. However, they will reconnect too, if needed.

It would help troubleshooting if you could be more specific.
Has the modem actually disconnected itself or not?
Are both your PCs standalone?

To add a configuration I would check: is there any chance that you have disabled redialling? Or that perhaps some anti-malware program is blocking redial?  It is not completely crazy that security applications block redial, BTW.

If on the other hand, the connection just dies (no real disconnect/hangup), it would be worth while to check your firewall settings. During a connection signals are sent "across the wire" to keep the connection alive, and what if these signals cannot get out?