This paper presents a method of modelling and analysing asynchronous traffic mode (ATM) networks, i.e. we study the performance characteristics of different ATM systems where the input is an aggregate cell arrival stream resulting from the superposition of different ATM traffic sources with different traffic characteristics. ATM traffic sources may range from constant bit rate sources (CBR) to variable bit rate sources (VBR). Flexible statistical multiplexing schemes must be adopted in order for the ATM networks to handle the traffic sources more efficiently, and to increase the bandwidth utilization as much as possible. In particular, we investigate two ATM traffic modelling approaches: the on-off model, and the Markov Modulated Poisson Process (MMPP) model. The performance analysis gives as output analytical formulas for the cell waiting time distribution and the cell loss probability in an ATM multiplexer queue. The multiplexer queue may represent a single ATM switching node in which the aggregate cell arrival stream arrives.