This paper proposes a new model which can approximate the delay time distribution in the Internet. It is well known that the delay time in communication links follows the exponential distribution. However, the earlier models cannot explain the distribution when a communication link is heavily overloaded. This paper proposes to use the M/M/S(m) model for the Internet. We have applied our model to the measurement results. This paper deals with one-way delay because it reflects the actual characteristics of communication links. Most measurement statistics in the Internet have been based on round-trip time delay between two end nodes. These characteristics are easily measured by sending sample packets from one node to the other. The receiver side echoes back the packets. However, the results are not always useful. A long distance communication link, such as a leased line, has two different fibers or wires for each direction: an incoming link, and an outgoing link. When the link is overloaded, the traffic in each link is quite different. The measurement of one-way delay is especially important for multimedia communications, because audio and video transmissions are essentially one-way traffic.