Information Transmission over the Postal Channel with and without Feedback

The postal channel models a postal system in which letters, each consisting of a number of characters, are sometimes lost. We study the postal channel with variable-length letters and variable-length coding over letters, both with and without letter-by-letter feedback. Without allowing letter lengths to encode information, we examine one feedback strategy consisting of automatic repeat-request (ARQ) with exponentially increasing letter lengths. For this strategy we investigate an alternative notion of information rate per character, based upon the total, random number of characters required to convey the messages instead of its expectation. This information rate exhibits a phase transition in its convergence as the number of messages becomes large: if the letter lengths increase by a factor less than the inverse of the probability that a letter is lost, it converges to the channel capacity; otherwise, it converges to a number strictly larger than channel capacity. More generally, when we allow both the characters and the length of a letter to convey information, we compute the corresponding channel capacity with and without feedback, and find that it is twice the channel capacity of the original postal channel without allowing letter lengths to encode information