  We analyze the path arrival rate for an inroom radio channel with directive
antennas. The impulse response of this channel exhibits a transition from early
separate components followed by a diffuse reverberation tail. Under the
assumption that the transmitter's (or receiver's) position and orientation are
picked uniformly at random we derive an exact expression of the mean arrival
rate for a rectangular room predicted by the mirror source theory. The rate is
quadratic in delay, inversely proportional to the room volume, and proportional
to the product of beam coverage fractions of the transmitter and receiver
antennas. Making use of the exact formula, we characterize the onset of the
diffuse tail by defining a "mixing time" as the point in time where the arrival
rate exceeds one component per transmit pulse duration. We also give an
approximation for the power-delay spectrum. It turns out that the power-delay
spectrum is unaffected by the antenna directivity. However, Monte Carlo
simulations show that antenna directivity does indeed play an important role
for the distribution of instantaneous mean delay and rms delay spread
