Degradedness and Secrecy in Memoryless Queues

We study exponential-server timing channels, in which the transmitter encodes a message using a sequence of packet inter-arrival times and the receiver observes the corresponding inter-departure times from an exponential-server queue. We show that if the service rate of one such queue is faster than that of another, then the latter channel is stochastically degraded with respect to the former. This combined with Burke's theorem implies that the slower queue is more entropy increasing than the first. These facts put together provide a converse for the secrecy capacity of the wiretapped version of the channel, matching an earlier achievability result in the literature.