Over time, cyber-physical systems are becoming mixed criticality systems. As the complexity and the size of these systems grow, computation/communication resources should be more efficient than with traditional systems. TTEthernet is a communication infrastructure that enables the use of a single physical communication infrastructure for distributed mixed criticality applications while providing timely determinism. TTEthernet distinguishes between two traffic categories: the standard event triggered and the time triggered. The latter, for which higher priority is granted, is subject to strong timing guarantees because of strict periodicity constraint that fixes start-time cycles of time-triggered messages. In addition, event-triggered traffic includes rate-constrained messages that are of lower priority and have a minimum time interval between their transmission. The paper proposes both an on-line efficient scheduling algorithm and a necessary and sufficient schedulability condition based on the worst case response time computation for rate-constrained messages while taking into account time-triggered messages transmission.