|There's more than one way to do things|
So, if I understand correctly, you have a flat rate for your internet access and want to make sure your box is always on. And probably your ISP disconnects you every once in a while. Right?
Well then, if you are disconnected after a certain idle time, you could prevent that by sending out a ping (one packet is enough) every 60 second or so.
Usually, you are disconnected anyway after a certain time, wether your link is active or not (T-Online does it after 24h, I believe, AOL even earlier; don't know for sure, though, cause I use other ISP's). In that case, you can want to reconnect.
Obviously you're using Unix. The following refers to Linux but should be valid for other flavours, too. Look for appropiate commands there.
Checking for a link can be done by checking wether or not the modem device exists, usually named ppp0. Check with ifconfig to see all devices or ifconfig ppp0 to check ppp0 directly.
To restart automagically, use the diald. A simple ping (or any network action) will cause your box to dial in.
Hope that helps ...