A method that seeks to provide a satisfactory quality of service for a stream of packets through a network. The illustrative embodiment of the present invention seeks to provide a satisfactory quality of service for a stream of packets by periodically or sporadically evaluating one or more alternative paths through the network and by sending the packets through a path with an acceptable quality of service. Normally, neither the source node nor any other node in a packet's path controls its route after it has left the node. In contrast, the illustrative embodiment of the present invention has the capability to affect the packet's path through the network by sending the packet to an intermediate or “ricochet” node in the network, which forwards the packet to the destination node. In effect, the source node can, if it deems appropriate, ricochet the packet off of the intermediate node rather than allowing the packet to take its normal direct path through the network.