This paper describes a new network scheme that minimizes the consumption of network resources for on-demand streaming-video distribution through multicasting. In general, on-demand video distribution has been realized through unicasting. Asynchronous multicast techniques that reduce the load on the on-demand video server have been studied, but there has been little consideration of ways to reduce the network load. We have developed a statistical traffic-control theory based on asynchronous multicasting that can be used to adjust traffic on the trunk and branch links along the distribution tree to meet the network bandwidth design requirements to ensure QOS. In addition, we propose a dynamic bandwidth-allocation and traffic-adaptation algorithm that allows finite resources to be shared among different video deliveries by assigning the available bandwidth for each according to request rates. To show the implementability of these ideas, we consider their operation with protocols that autonomously construct a multicast tree and guarantee the QOS. We also discuss the effects of using this scheme based on a numerical analysis and simulation results regarding real traffic traces and the service-blocking rate.