Both deadlock avoidance and recovery techniques suffer from severe performance degradation when the network is close to or beyond saturation. This performance degradation appears because messages block in the network faster than they are drained by the escape paths in the deadlock avoidance strategies or the deadlock recovery mechanism.Many parallel applications produce bursty traffic that may saturate the network during some intervals [14,8], significantly increasing execution time. Therefore, the use of techniques that prevent network saturation are of crucial importance. Although several mechanisms have been proposed in the literature to reach this goal, some of them introduce some penalty when the network is not fully saturated, require complex hardware to be implemented or do not behave well under all network load conditions. In this paper, we propose a new mechanism to avoid network saturation that overcomes these drawbacks.