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In this paper, we present an analytical study for the performance of message dissemination in vehicular ad hoc networks (VANETs) with two priority classes. It is assumed that the message traffic generated by event-driven safety applications has higher priority compared to the remaining network traffic. First, we derive the distribution of the number of concurrent transmissions of lower priority messages in the system at the steady state, through a birth-death process analysis. The distribution has a simple product form solution. We also derive the percentage of destination node population which cannot receive the message error free due to interference. Subsequently, we determine the average forwarding distance and the number of nodes which receive a high-priority message in the presence of low-priority traffic. Numerical results are provided along with simulation results that confirm the accuracy of the proposed analysis. The distribution of the number of concurrent transmissions is shown to have a bell-shape curve. Results also show that larger transmission ranges do not necessarily improve the per hop safety-message forwarding distance as more nodes may be exposed to interference, especially in networks with higher node densities.