In this paper, we propose some representations of a generalized linear birth process called fractional Yule process (fYp). We also derive the probability distributions of the random birth and sojourn times. The inter-birth time distribution and the representations then yield algorithms on how to simulate sample paths of the fYp. We also attempt to estimate the model parameters in order for the fYp to be usable in practice. The estimation procedure is then tested using simulated data as well. We also illustrate some major characteristics of fYp which will be helpful for real applications.