The paper contributes to the teletraffic analysis and quality of service of packet transmission of peer-to-peer systems. We look at the packet transmission processes that are induced by a peer-to-peer overlay disseminating real-time traffic from a statistical perspective. Applying the theory of extreme events, we derive that the asymptotic distribution of the end-to-end packet delay (i.e. packet delivery time between sender and receiver peers) is subexponential heavy-tailed if the inter-arrival times between packets are regularly varying heavy-tailed distributed. To evaluate the probability to miss a packet as a quality-of-service attribute one requires a set of observable variables based on the realized flows such as the end-to-end packet delays, the packet inter-arrival times and packet lengths of single or aggregated streams associated with a peer.