An example method is provided and includes receiving a relay address allocation request from an endpoint, the relay address allocation request comprises a unique session identifier that identifies a conference session joined by the endpoint for media streaming; determining a relay candidate comprising a relay transport address for allocating to each endpoint of the conference session having the unique session identifier. Further, the method includes mapping the relay candidate with the unique session identifier and sending a relay address allocation response that comprises at least the relay candidate mapped with the unique session identifier. The method further includes receiving a single copy of one or more media stream packets from the conference controller and relaying the one or more media stream packets via the relay transport address identified by the unique session identifier to each of the one or more endpoints in the session having the unique session identifier.