First step is to check whether your computers can see the local router at your end, i.e. ping the local gateway.
Then do a tracert to www.thinkbroadband.com
and see at what point it starts to lose the data. If the problem was with the downlink from satellite and break out to the internet one would normally expect to see more users affected.
One possibility is that the issue is simply congestion, and that rebooting just gives enough time for a spike of traffic on the service to vanish.