The M/PH/∞ system is introduced in this paper to analyze the superposition of a large number of data connections on an asynchronous transfer mode (ATM) link. In this model, information is transmitted in bursts of data arriving at the link as a Poisson process of rate λ and burst durations are PH distributed with unit mean. Some transient characteristics of the M/PH/∞ system, namely the duration θ of an excursion by the occupation process {Xt} above the link transmission capacity C, the area V swept under process {Xt} above C and the number of customers arriving in such an excursion period, are introduced as performance measures. Explicit methods of computing their distributions are described. It is then shown that, as conjectured in earlier studies, random variables Cθ, CV, and N converge in distribution as C tends to infinity while the utilization factor of the link defined by γ = λ/C is fixed in (0,1), towards some transient characteristics of an M/M/1 queue with input rate γ and unit service rate. Further simulation results show that after adjustment of the load of the M/M/1 queue, a similar convergence result holds for the superposition of a large number of On/Off sources with various On and Off period distributions. This shows that some transient quantities associated with an M/M/1 queue can be used in the characterization of open loop multiplexing of a large number of On/Off sources on an ATM link.
