  Multi-user diversity is considered when the number of users in the system is
random. The complete monotonicity of the error rate as a function of the
(deterministic) number of users is established and it is proved that
randomization of the number of users always leads to deterioration of average
system performance at any average SNR. Further, using stochastic ordering
theory, a framework for comparison of system performance for different user
distributions is provided. For Poisson distributed users, the difference in
error rate of the random and deterministic number of users cases is shown to
asymptotically approach zero as the average number of users goes to infinity
for any fixed average SNR. In contrast, for a finite average number of users
and high SNR, it is found that randomization of the number of users
deteriorates performance significantly, and the diversity order under fading is
dominated by the smallest possible number of users. For Poisson distributed
users communicating over Rayleigh faded channels, further closed-form results
are provided for average error rate, and the asymptotic scaling law for ergodic
capacity is also provided. Simulation results are provided to corroborate our
analytical findings.
