  We consider the problem of stabilizing an undisturbed, scalar, linear system
over a "timing" channel, namely a channel where information is communicated
through the timestamps of the transmitted symbols. Each symbol transmitted from
a sensor to a controller in a closed-loop system is received subject to some to
random delay. The sensor can encode messages in the waiting times between
successive transmissions and the controller must decode them from the
inter-reception times of successive symbols. This set-up is analogous to a
telephone system where a transmitter signals a phone call to a receiver through
a "ring" and, after the random delay required to establish the connection; the
receiver is aware of the "ring" being received. Since there is no data payload
exchange between the sensor and the controller, this set-up provides an
abstraction for performing event-triggering control with zero-payload rate. We
show the following requirement for stabilization: for the state of the system
to converge to zero in probability, the timing capacity of the channel should
be, essentially, at least as large as the entropy rate of the system.
Conversely, in the case the symbol delays are exponentially distributed, we
show an "almost" tight sufficient condition using a coding strategy that
refines the estimate of the decoded message every time a new symbol is
received. Our results generalize previous zero-payload event-triggering control
strategies, revealing a fundamental limit in using timing information for
stabilization, independent of any transmission strategy.
