Real-Time communication requires a performance guarantee from the underlying network. In order to analyse the network performance, we must find the traffic characterization in every server of the network. Due to the strong experimental evidence that network traffic is self-similar in nature, it is important to study the problem to see whether the service of a server changes the self-similarity property of the input traffic. We establish a model of a single server with an infinite buffer and prove that if the queue length has finite second-order moment then the input process being strong asymptotically second-order self-similar (sas-s) is equivalent to the output process also bearing the sas-s property. Given the method for determinating the worst case cell delay for an ATM switch with self-similar input traffic, with this proof we can determine the end-To-end delay for much real-Time communications in an ATM network by summing the cell delay experienced by each of the ATM switches in the connection.