A state space model for a class of Large Homogeneous communication networks with Markovian access control disciplines is considered. It is assumed that each user has a finite or infinite buffer. Asymptotic analysis of the resulting stochastic finite-difference equations reveals the following, previously unknown properties common to all networks in this class: (i) the load line is an algebraic curve of order N, where N is the length of the buffers; (ii) for N = ∞, the average steady state buffer occupancy is yS = y1 (1 - y1) where y1 is the intersection point of the load and transmission lines; an analogous expression is derived also for N < ∞; (iii) the local stability of a steady state is determined by the slopes of the load and transmission lines. Numerical experiments are described to support these findings. The results of this paper give a solution to the problem of dynamic analysis of Large Homogeneous communication networks with Markovian access control disciplines if the steady state is unique. The case of multiple steady states is considered in Part II.