Multi-User Diversity with Random Number of Users

Abstract: 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.