In this paper, we derive the value of a signal obtained by a traveller prior to the choice of departure time. The signal does not have to be a perfect prediction of the travel time. It is sufficient that it carries information about the travel time. The traveller may then consider the distribution of travel time conditional on the signal to increase her expected utility by making a better informed choice of departure time. We show that a signal always increases the expected utility compared to the situation without any signal. For a broad class of signals, the expected utility is monotone with respect to signal strength. Furthermore, we demonstrate that even perfect travel time information does not necessarily eliminate the cost of travel time variability and we establish necessary and sufficient conditions for when it does. We find that the predictable part of travel time variability may or may not be costly, depending on the shape of the traveller's scheduling utility at the origin of the trip. Using estimates of scheduling preferences from the literature, we show that the cost of predictable travel time variability may constitute a substantial part of the total cost of travel time variability. In a particular case of scheduling preferences, travel time distribution and noise distribution, we establish an analytic relationship between the strength of the signal and the expected utility of the trip and evaluate the marginal cost of signal weakness. This knowledge may facilitate design and cost-benefit analysis of traveller information systems and policies decreasing travel time variability.