We obtain upper bounds on the probability of buffer overflow for an ATM multiplexer of L identical packetized voice sources. The multiplexer is modelled by a FCFS single server queue. The arrivals at the multiplexer are a homogenous superposition of the arrivals from L independent identical sources, with each source modelled by a copy of a discrete time Markov Chain which we call the Cell Level Model. Throughout, appropriate parameters are scaled with L, to maintain a constant load over all superposition sizes. The probability that, the queue-length (q^) of the queue in a finite buffer exceeds the buffer size b, is bounded above by the probability that the queue-length (qL) of the queue m an infinite buffer exceeds length b In order to bound the former above, we find upper bounds or approximations for the latter by using the theory of, • Large Deviations, to determine its asymptotics for large b, • Martingales, to obtain upper bounds, valid for all positive b, • Large Deviations, to determine its asymptotics for large L for time rescaled (proportional to L) arrival processes. These demonstrate the multiplexing gam and economies of scale obtainable from large and small buffers and large multiplexers, respectively.