Finite Block-Length Achievable Rates for Queuing Timing Channels

The exponential server timing channel is known to be the simplest, and in some sense canonical, queuing timing channel. The capacity of this infinite-memory channel is known. Here, we discuss practical finite-length restrictions on the codewords and attempt to understand the amount of maximal rate that can be achieved for a target error probability. By using Markov chain analysis, we prove a lower bound on the maximal channel coding rate achievable at blocklength $n$ and error probability $\epsilon$ is approximated by $C- n^{-1/2} \sigma Q^{-1}(\epsilon)$ where $Q$ denotes the Q-function and $\sigma^2$ is the asymptotic variance of the underlying Markov chain. A closed form expression for $\sigma^2$ is given.