  This work considers an asynchronous $\textsf{K}_\text{a}$-active-user
unsourced multiple access channel (AUMAC) with the worst-case asynchronicity.
The transmitted messages must be decoded within $n$ channel uses, while some
codewords are not completely received due to asynchronicities. We consider a
constraint of the largest allowed delay of the transmission. The AUMAC lacks
the permutation-invariant property of the synchronous UMAC since different
permutations of the same codewords with a fixed asynchronicity are
distinguishable. Hence, the analyses require calculating all
$2^{\textsf{K}_\text{a}}-1$ combinations of erroneously decoded messages.
Moreover, transmitters cannot adapt the corresponding codebooks according to
asynchronicity due to a lack of information on asynchronicities. To overcome
this challenge, a uniform bound of the per-user probability of error (PUPE) is
derived by investigating the worst-case of the asynchronous patterns with the
delay constraint. Numerical results show the trade-off between the
energy-per-bit and the number of active users for different delay constraints.
In addition, although the asynchronous transmission reduces interference, the
required energy-per-bit increases as the receiver decodes with incompletely
received codewords, compared to the synchronous case.
