We study the communication complexity of asynchronous distributed algorithms. Such algorithms can generate excessively many messages in the worst case. Nevertheless, we show that, under certain probabilistic assumptions, the expected number of messages generated per time unit is bounded by a polynomial function of the number of processors under a very general model of distributed computation. Furthermore, for constant-degree processor graphs, the expected number of generated messages is only O(nT) , where n is the number of processors and T is the running time. We conclude that (under our model) any asynchronous algorithm with good time complexity will also have good communication complexity, on the average.