IMO, your modem going to sleep is very unlikely. You could of course easily confirm or refute that theory by inviting a friend to connect while you're at home and using the Internet. Or, leave your computer on and e.g. running a ping, then go to an outside location to test.
AFAIK, Gateway (the computer manufacturer) does not make and has never made modems. However, some modem/router combinations are called "gateways". If you post the make and model of the modem, you may get better help.
Possibly, your 350 was not properly set up for Internet viewing, or (if your modem/router requires it), you didn't set up the port forwarding correctly.
When playing a video from your home computer, hold down the Ctrl and Shift keys and left-click your mouse in the video window. A BoxInfo window should appear. Confirm that WAN Access and Register are both Yes.
If you get that far and are still having trouble, copy the ID from the BoxInfo window and remove the four dashes. You should have 32 letters and numbers left. Then, visit
http://sparcs.slingmedia.com/getDeviceInfo2.asp?id=12345678901234567890123456789012 , replacing the digits after id= with your ID. You'll see a report of the capability to connect to your box in various ways. Ideally, you should see (among other things):
Capability=Supported and Passed
If so, then the 'road' connection problem is probably at the viewing end. For example, you may not be able to connect from work, because streaming is blocked by the corporate firewall. Confirm this by trying to connect e.g. from a friend's home.
Thanks for posting in the forums! I'm sorry to hear you are having trouble connecting outside of your network! The above response is a very great suggestion. I would just like to add an additional article that may also assist you.
Here is the link:
Hope this helps!
The Sling Moderation Team