An additive exponential noise channel with a transmission deadline

We derive the maximum mutual information for an additive exponential noise (AEN) channel with a peak input constraint. We find that the optimizing input density is mixed (with singularities) similar to previous results for AEN channels with a mean input constraint. Likewise, the maximum mutual information takes a similar form, though obviously the maximum for the peak constraint is smaller than for the corresponding mean-constrained channel. This model is inspired by multiple biological phenomena and processes which can be abstracted as follows: inscribed matter is sent by an emitter, moves through a medium, and arrives eventually at its destination receptor. The inscribed matter can convey information in a variety of ways such as the number of signaling quanta - molecules, macromolecular complexes, organelles, cells and tissues - that are emitted as well as the detailed pattern of their release. However, rather than focus on a general class of emitter-receptor systems or a particular exemplar of biomedical importance, our ultimate goal is to provide bounds on the potential efficacy of timed-release signaling for any system which emits identical signaling quanta. That is, we seek to apply one of the most potent aspects of information theory to biological signaling - mechanism blindness - in the hopes of gaining insights applicable to diverse systems that span a wide range of spatiotemporal scales.