We analyze the benefits of an automatic repeat-request (AKOl scheme employed in doped belief-propagation decoding for increasing the throughput of fountain encoded data transmissions. The proposed doping mechanism selects doping symbols randomly from the set of input symbols contributing to degree-two output symbols. Here an output symbol is the encoded symbol whose code-graph links to input symbols decoded thus far have been severed. This doping approach always ensures releasing of at least one output symbol, thus increasing the number of degree-one output symbols (the ripple). Using a random walk analysis, we study the belief propagation decoding with degree-two random doping for a fountain code with symbols drawn from an Heal Soliton distribution. We show that the decoding process is a renewal process whereas the process starts all over afresh after each doping. The approximate interdoping process analysis revolves around a random walk model for the ripple size. We model the sequence of the ripple size increments (due to doping and/or decoding) as an iid sequence of shifted and truncated Poisson random variables. This model furnishes a prediction on the number of required doping symbols and, furthermore, the ARQ throughput cost analysis. We also find that the Ideal Soliton significantly outperforms the Robust Soliton distribution in our ARQ-doping scheme.