  In multi-class communication networks, traffic surges due to one class of
users can significantly degrade the performance for other classes. During these
transient periods, it is thus of crucial importance to implement priority
mechanisms that conserve the quality of service experienced by the affected
classes, while ensuring that the temporarily unstable class is not entirely
neglected. In this paper, we examine the complex interaction occurring between
several classes of traffic when classes obtain bandwidth proportionally to
their incoming traffic.
  We characterize the evolution of the network from the moment the initial
surge takes place until the system reaches its equilibrium. Using an
appropriate scaling, we show that the trajectories of the temporarily unstable
class can be described by a differential equation, while those of the stable
classes retain their stochastic nature. A stochastic averaging phenomenon
occurs and the dynamics of the temporarily unstable and the stable classes
continue to influence one another. We further proceed to characterize the
obtained differential equations and the stability region under this scaling for
monotone networks. We illustrate these result on several toy examples and we
finally build a penalization rule using these results for a network integrating
streaming and elastic traffic.
