Optimal Age over Erasure Channels

Given a source that produces a letter every T-s seconds and an erasure channel that can be used every T-c seconds, we ask what is the coding strategy that minimizes the time-average "age of information" that an observer of the channel output incurs. We will see that one has to distinguish the cases when the source and channel-input alphabets have equal or different size. In the first case, we show that a trivial coding strategy is optimal and a closed form expression for the age may be derived. In the second, we use random coding argument to bound the average age and show that the average age achieved using random codes converges to the optimal average age as the source alphabet becomes large.