Delays and the Capacity of Continuous-time Channels

Any physical channel of communication offers two potential reasons why its capacity (the number of bits it can transmit in a unit of time) might be unbounded: (1) Infinitely many choices of signal strength at any given instant of time, and (2) Infinitely many instances of time at which signals may be sent. However channel noise cancels out the potential unboundedness of the first aspect, leaving typical channels with only a finite capacity per instant of time. The latter source of infinity seems less studied. A potential source of unreliability that might restrict the capacity also from the second aspect is delay: Signals transmitted by the sender at a given point of time may not be received with a predictable delay at the receiving end. Here we examine this source of uncertainty by considering a simple discrete model of delay errors. In our model the communicating parties get to subdivide time as microscopically finely as they wish, but still have to cope with communication delays that are macroscopic and variable. The continuous process becomes the limit of our process as the time subdivision becomes infinitesimal. We taxonomize this class of communication channels based on whether the delays and noise are stochastic or adversarial; and based on how much information each aspect has about the other when introducing its errors. We analyze the limits of such channels and reach somewhat surprising conclusions: The capacity of a physical channel is finitely bounded only if at least one of the two sources of error (signal noise or delay noise) is adversarial. In particular the capacity is finitely bounded only if the delay is adversarial, or the noise is adversarial and acts with knowledge of the stochastic delay. If both error sources are stochastic, or if the noise is adversarial and independent of the stochastic delay, then the capacity of the associated physical channel is infinite.