  A single primary user cognitive radio system with multi-user diversity at the
secondary users is considered where there is an interference constraint between
secondary and primary users. The secondary user with the highest instantaneous
SNR is selected for communication from a set of active users which also
satisfies the interference constraint. The active number of secondary users is
shown to be binomial, negative binomial, or Poisson-binomial distributed
depending on various modes of operation. Outage probability in the slow fading
scenario is also studied. This is then followed by a derivation of the scaling
law of the ergodic capacity and BER averaged across the fading, and user
distribution for a large mean number of users. The ergodic capacity and average
BER under the binomial user distribution is shown to outperform the negative
binomial case with the same mean number of users. Moreover, the Poisson
distribution is used to approximate the user distribution under the non-i.i.d
interference scenario, and compared with binomial and negative binomial
distributions in a stochastic ordering sense. Monte-Carlo simulations are used
to supplement our analytical results and compare the performances under
different user distributions.
