  From an information theoretic perspective, joint communication and sensing
(JCAS) represents a natural generalization of communication network
functionality. However, it requires the re-evaluation of network performance
from a multi-objective perspective. We develop a novel mathematical framework
for characterizing the sensing and communication coverage probability and
ergodic rate in JCAS networks. We employ a formulation of sensing parameter
estimation based on mutual information to extend the notions of coverage
probability and ergodic rate to the radar setting. We define sensing coverage
probability as the probability that the rate of information extracted about the
parameters of interest associated with a typical radar target exceeds some
threshold, and sensing ergodic rate as the spatial average of the
aforementioned rate of information. Using this framework, we analyze the
downlink sensing and communication coverage and rate of a mmWave JCAS network
employing a shared waveform, directional beamforming, and monostatic sensing.
Leveraging tools from stochastic geometry, we derive upper and lower bounds for
these quantities. We also develop several general technical results including:
i) a generic method for obtaining closed form upper and lower bounds on the
Laplace Transform of a shot noise process, ii) a new analog of H{\"o}lder's
Inequality to the setting of harmonic means, and iii) a relation between the
Laplace and Mellin Transforms of a non-negative random variable. We use the
derived bounds to numerically investigate the performance of JCAS networks
under varying base station and blockage density. Among several insights, our
numerical analysis indicates that network densification improves sensing SINR
performance -- in contrast to communications.
