Analog lines can handle speeds up to 28.8 kbps. This is the same grade as your typical home phone line. You probably don't want one of these. The cost of a digital line depends on the distance between you and your service provider. An alternative to digital dedicated lines is frame relay.
Frame relay is the new technology on the block.
Not all service providers support frame relay. Check with your service provider. For the purposes of this article, I will assume you are going to go with a digital line at 56 kbps. This is the most common Internet connection. With a dedicated connection, your Linux box is available 24 hours a day to access the Internet.