Sparse telephone gateway for internet telephony

Internet telephony is promising for long-distance calls. To serve the general public, a service provider can operate a telephone gateway in each servicing city to bridge the local telephone network and the Internet, so that users can use telephones or mobile phones to access these telephone gateways for long-distance calls. The service provider should serve many cities to attain good service coverage but it is costly to operate telephone gateways in all of these cities. In this paper, we propose a sparse telephone gateway configuration to serve many cities cost-effectively. Using this configuration, if the service provider serves N cities, it operates telephone gateways in only M   cities where M