Venkat Anantharam The Reliability of Communicating Information Through Timing via an Exponential Server

A single server queue can be viewed as a communication channel where information is encoded into the times of arrivals of packets into the queue and needs to be decoded from the departure times. Such channels are of interest in computer security, since their existence provides a way to leak information across security boundaries in a covert way. Anantharam and Verdu (1994) determined that the Shannon capacity of the timing channel associated to an exponential server of rate mu is mu/e nats per unit time. Recently Arikan (2002) studied the reliability of this channel, i.e. the exponent of the average error probability of communicating over this channel (the asymptotics being in the time over which one uses the channel) as a function of the data rate (between zero and the Shannon capacity). He found upper and lower bounds that match above (μ log 2)/4. Also, remarkably, the reliability exponent is identical, over the region it has been determined by Arikan, to that of the well studied ideal photon channel with bound mu on the peak input rate : in this channel coding is done by modulating the intensity of a Poisson process and decoding is done from the observed count of the points (this channel also has Shannon capacity μ/e nats per unit time). An open question was whether the queueing channel and the photon count channel are equally reliable. We resolve this question by showing that the queueing channel is strictly more reliable than the photon channel for small data rates. We prove that its reliability exponent at zero (vanishingly small) rate is μ/2, while that of the photon channel is known since Wyner (1988) to be μ/4. The proof involves some interesting point process mathematics that is likely to be of wider interest to queueing theorists.