Zero-Error Capacity of a Class of Timing Channels

We analyze the problem of zero-error communication through timing channels that can be interpreted as discrete-time queues with bounded waiting times. The channel model includes the following assumptions: 1) Time is slotted, 2) at most $N$ "particles" are sent in each time slot, 3) every particle is delayed in the channel for a number of slots chosen randomly from the set $\{0, 1, \ldots, K\}$, and 4) the particles are identical. It is shown that the zero-error capacity of this channel is $\log r$, where $r$ is the unique positive real root of the polynomial $x^{K+1} - x^{K} - N$. Capacity-achieving codes are explicitly constructed, and a linear-time decoding algorithm for these codes devised. In the particular case $N = 1$, $K = 1$, the capacity is equal to $\log \phi$, where $\phi = (1 + \sqrt{5}) / 2$ is the golden ratio, and the constructed codes give another interpretation of the Fibonacci sequence.