Speech-based degree of sleepiness estimation is an emerging research problem. This paper investigates an end-to-end approach, where given raw waveform as input, a convolutional neural network (CNN) estimates at its output the degree of sleepiness. Within this approach, we investigate constraining the first layer processing and integration of speech production knowledge through transfer learning. We evaluate these methods on the continuous sleepiness corpus of the Interspeech 2019 Computational Paralinguistics (ComParE) Challenge and demonstrate that the proposed approach consistently yields competitive systems. In particular, we observe that integration of speech production knowledge aids in improving the performance and yields systems that are complementary.