  To account for the massive uncoordinated random access scenario, which is
relevant for the Internet of Things, Polyanskiy (2017) proposed a novel
formulation of the multiple-access problem, commonly referred to as unsourced
multiple access, where all users employ a common codebook and the receiver
decodes up to a permutation of the messages. We extend this seminal work to the
case where the number of active users is random and unknown a priori. We define
a random-access code accounting for both misdetection (MD) and false alarm
(FA), and derive a random-coding achievability bound for the Gaussian multiple
access channel. Our bound captures the fundamental trade-off between MD and FA.
It suggests that the lack of knowledge of the number of active users entails a
small penalty in energy efficiency when the target MD and FA probabilities are
high. However, as the target MD and FA probabilities decrease, the energy
efficiency penalty becomes significant. For example, in a typical IoT scenario,
the required energy per bit to achieve both MD and FA probabilities below 0.1,
predicted by our bound, is only 0.5-0.7 dB higher than that predicted by the
bound in Polyanskiy (2017) for a known number of active users. This gap
increases to 3-4 dB when the target MD and/or FA probability is 0.001. Taking
both MD and FA into account, we use our bound to benchmark the energy
efficiency of slotted ALOHA with multi-packet reception, of a decoder that
treats interference as noise, and of some recently proposed coding schemes.
Numerical results suggest that, when the target MD and FA probabilities are
high, it is effective to estimate the number of active users, then treat this
estimate as the true value, and use a coding scheme that performs well for the
case of known number of active users. However, this approach becomes energy
inefficient when the requirements on MD and FA probabilities are stringent.
