  In this paper, we consider a decentralized wireless communication network
with a fixed number $u$ of frequency sub-bands to be shared among $N$
transmitter-receiver pairs. It is assumed that the number of active users is a
random variable with a given probability mass function. Moreover, users are
unaware of each other's codebooks and hence, no multiuser detection is
possible. We propose a randomized Frequency Hopping (FH) scheme in which each
transmitter randomly hops over a subset of $u$ sub-bands from transmission to
transmission. We derive lower and upper bounds on the mutual information of
each user and demonstrate that, for large Signal-to-Noise Ratio (SNR) values,
the two bounds coincide. This observation enables us to compute the sum
multiplexing gain of the system and obtain the optimum hopping strategy for
maximizing this quantity. We compare the performance of the FH system with that
of the Frequency Division (FD) system in terms of several performance measures
and show that (depending on the probability mass function of the number of
active users) the FH system can offer a significant improvement implying a more
efficient usage of the spectrum.
