Delay on Broadcast Erasure Channels Under Random Linear Combinations

We consider a transmitter broadcasting random linear combinations (over a field of size d) formed from a block of c packets to a collection of n receivers, where the channels between the transmitter and each receiver are independent erasure channels with reception probabilities q = (q
₁
,..., q
_n
). We establish several properties of the random delay until all n receivers have recovered all c packets, denoted Y
_n:n
^(c)
. First, we provide lower and upper bounds, exact expressions, and a recurrence for the moments of Y
_n:n
^(c)
. Second, we study the delay per packet Y
_n:n
^(c)
/c as a function of c, including the asymptotic delay (as c → ∞) and monotonicity (in c) properties of the delay per packet. Third, we employ extreme value theory to investigate Y
_n:n
^(c)
as a function of n (as n → ∞). Several results are new, some results are extensions of existing results, and some results are the proofs of known results using new (probabilistic) proof techniques.