This paper focuses on energy-efficient longitudinal controller design for a connected automated truck that travels in mixed traffic consisting of connected and non-connected vehicles. The truck has access to information about connected vehicles beyond line of sight using vehicle-to-everything (V2X) communication. A novel connected cruise control design is proposed which incorporates additional delays into the control law when responding to distant connected vehicles to account for the finite propagation speed of traffic waves. The speeds of non-connected vehicles are modeled as stochastic processes. A fundamental theorem is proven which links the spectral properties of the motion signals to the average energy consumption. Controller synthesis for gain parameters is conducted over downstream traffic data and evaluated over a combination of synthetic and real cycles. It is demonstrated that even with lean penetration of connected vehicles, our controller can bring significant energy savings.