We propose a probabilistic network model, called asynchronous bounded expected delay (ABE), which requires a known bound on the expected message delay. In ABE networks all asynchronous executions are possible, but executions with extremely long delays are less probable. Thus, the ABE model captures asynchrony that occurs in sensor networks and ad-hoc networks. At the example of an election algorithm, we show that the minimal assumptions of ABE networks are sufficient for the development of efficient algorithms. For anonymous, unidirectional ABE rings of known size n we devise a probabilistic election algorithm having average message and time complexity O(n). © 2011 Elsevier B.V.