Zero-Error Capacity of a Class of Timing Channels

Abstract: 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.