On information transmission over a finite buffer channel
We study information transmission through a finite buffer queue. We model the channel as a finite-state channel whose state is given by the buffer occupancy upon packet arrival; a loss occurs when a packet arrives to a full queue. We study this problem in two contexts: one where the state of the buffer is known at the receiver, and the other where it is unknown. In the former case, we show that the capacity of the channel depends on the long-term loss probability of the buffer. Thus, even though the channel itself has memory, the capacity depends only on the stationary loss probability of the buffer. The main focus of this correspondence is on the latter case. When the receiver does not know the buffer state, this leads to the study of deletion channels, where symbols are randomly dropped and a subsequence of the transmitted symbols is received. In deletion channels, unlike erasure channels, there is no side-information about which symbols are dropped. We study the achievable rate for deletion channels, and focus our attention on simple (mismatched) decoding schemes. We show that even with simple decoding schemes, with independent and identically distributed (i.i.d.) input codebooks, the achievable rate in deletion channels differs from that of erasure channels by at most H-0(p(d)) - p(d) log (K-1)/(K) bits, for p(d) < 1 - K-1, where p(d) is the deletion probability, K is the alphabet size, and H-0((.)) is the binary, entropy function. Therefore, the difference in transmission rates between the erasure and deletion channels is not large for reasonable alphabet sizes. We also develop sharper lower bounds with the simple decoding framework for the deletion channel by analyzing it for Markovian codebooks. Here, it is shown that the difference between the deletion and erasure capacities is even smaller than that with i.i.d. input codebooks and for a larger range of deletion probabilities. We also examine the noisy deletion channel where a deletion channel is cascaded with a symmetric discrete memoryless channel (DMC). We derive a single letter expression for an achievable rate for such channels. For the binary case, we show that this result simplifies to max(0,1 - [H-0(theta) + theta H-0(p(e))]) where p(e) is the cross-over probability for the binary symmetric channel.
deletion channels erasure channels common subsequences channel capacity