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This paper focuses on understanding the scale and the distribution of "state overhead'' (briefly load) that is incurred on the routers by various value-added network services, e.g., IP multicast and IP traceback. This understanding is essential to developing appropriate mechanisms and provisioning resources so that the Internet can support such value-added services in an efficient and scalable manner. We mainly consider the number of end-to-end paths or trees intersecting at a router to represent the amount of state overhead at that router. Hence, we analyze the router-level intersection characteristics of end-to-end Internet paths or trees to approximate the state overhead distribution in the Internet. For the reliability of our analysis, a representative, end-to-end router-level Internet map is essential. Although several maps are available, they are at best insufficient for our analysis. Therefore, in the first part of our work, we exert a measurement study to obtain a large size end-to-end router-level map conforming to our constraints. In the second part, we conduct various experiments using our map and shed some light on the scale and distribution of state overhead of value-added Internet services in both unicast and multicast environments.