Network coding broadcast delay on erasure channels

We consider a broadcast erasure channel where a single transmitter broadcasts to M receivers, where the channel between the transmitter and each receiver is an independent erasure channel with success probabilities q = (q₁, ..., q_M). Prior work on this model by Cogill, Shrader and Ephremides (2011) established bounds on the expected delays incurred in transmitting a collection of C packets to each of the M receivers using both scheduling and random linear network coding (RLNC). In this work we extend these results in several ways. First, we provide exact expressions and tight upper and lower bounds for the expected delay under uncoded transmissions (UT) for arbitrary q. Second, we provide a recurrence relationship for the expected delay under RLNC that allows one to compute the exact delay for any finite C. Third, we prove convergence in probability and in rth mean of the normalized expected delay under RLNC as C → ∞ to 1/min_j q_j, with the implication that the normalized delay is solely a function of the quality of the worst erasure channel. Finally, we note our results are of general interest in probability theory as they address the moments of the maximum of (sums of) independent geometric random variables (UT, RLNC) and the convergence of moments of functions of sums of independent geometric random variables (RLNC).