The $M|G|infty$ busy server process provides a class of structural models for communication network traffic. In this dissertation, we study the asymptotic behavior of a network multiplexer, modeled as a discrete-time queue, driven by an $M|G|infty$ correlated arrival stream.

The asymptotic regimes considered here are those of heavy and light traffic. In heavy traffic, we show that the arising limits are described in terms of the classical Brownian motion and the $alpha$--stable L'{e}vy motion, under short- and long-range dependence, respectively. Salient features are then effectively captured by the exponential distribution and the Mittag-Leffler special function.

In light traffic, the analysis reveals the effect of two aspects of the $M|G|infty$ process, i.e., the session duration distribution $G$ and the gradual nature of the arrivals, as opposed to the instantaneous inputs of a standard $GI|GI|1$ queue. We exploit these asymptotic results to construct interpolation approximations for system quantities of interest, applicable to all traffic intensities.