In this paper we investigate connected cruise control in which vehicles rely on ad hoc wireless vehicle-to-vehicle communication to control their longitudinal motion. Intermittencies and packet drops in communication channels are shown to introduce stochastic delays in the feedback loops. Sufficient conditions for almost sure stability of equilibria are derived by analyzing the mean and covariance dynamics. In addition, the concept of nσ string stability is proposed to characterize the input-output response in steady state. The stability results are summarized using stability charts in the plane of the control gains and we demonstrate that the stable regimes shrink when the sampling time or the packet drop ratio increases. The mathematical tools developed allow us to design controllers that can achieve plant stability and string stability in connected vehicle systems despite the presence of stochastically varying delays in the control loop.