Signaling with identical tokens: Upper bounds with energy constraints

As system sizes shrink to the nanoscale, the usual macroscopic methods of communication using electromagnetic and acoustic waves become increasingly difficult and energy-inefficient owing to, essentially, a mismatch between realizable antenna sizes and the propagation characteristics of the medium. Thus, at the scale of microns and below, communication methods which utilize molecular messengers become increasingly attractive, a notion supported by the ubiquity of molecular signaling in biological systems, usually with identical molecules. In a large portion of previous work, time-varying signal molecule/token concentration is used as the observable and various analyses performed. However, from an information-theoretic standpoint, concentration masks the underlying process which consists, fundamentally, of signal token emission, transit through some medium, and reception. We build here on previous work to establish machinery which allows upper bounds to be derived on the identical token timing channel. We then consider the special case of exponential token transit times.