Scaling Laws for Gaussian Random Many-Access Channels

This paper considers a Gaussian multiple-access channel with random user activity where the total number of users
$\ell _{n}$
and the average number of active users
$k_{n}$
may grow with the blocklength
$n$
. For this channel, it studies the maximum number of bits that can be transmitted reliably per unit-energy as a function of
$\ell _{n}$
and
$k_{n}$
. When all users are active with probability one, i.e.,
$\ell _{n} = k_{n}$
, it is demonstrated that, if
$k_{n}$
is of an order strictly below
$n/\log n$
, then each user can achieve the single-user capacity per unit-energy
$(\log e)/N_{0}$
(where
$N_{0}/ 2$
is the noise power) by using an orthogonal-access scheme. In contrast, if
$k_{n}$
is of an order strictly above
$n/\log n$
, then the users cannot achieve any positive rate per unit-energy. Consequently, there is a sharp transition between orders of growth where interference-free communication is feasible and orders of growth where reliable communication at a positive rate per unit-energy is infeasible. It is further demonstrated that orthogonal-access schemes in combination with orthogonal codebooks, which achieve the capacity per unit-energy when the number of users is bounded, can be strictly suboptimal. When the user activity is random, i.e., when
$\ell _{n}$
and
$k_{n}$
are different, it is demonstrated that, if
$k_{n}\log \ell _{n}$
is sublinear in
$n$
, then each user can achieve the single-user capacity per unit-energy
$(\log e)/N_{0}$
. Conversely, if
$k_{n}\log \ell _{n}$
is superlinear in
$n$
, then the users cannot achieve any positive rate per unit-energy. Consequently, there is again a sharp transition between orders of growth where interference-free communication is feasible and orders of growth where reliable communication at a positive rate is infeasible that depends on the asymptotic behaviors of both
$\ell _{n}$
and
$k_{n}$
. It is further demonstrated that orthogonal-access schemes, which are optimal when all users are active with probability one, can be strictly suboptimal in general.