  In this paper, the trade-off between the number of transmissions (or
burstiness) $K_n=e^{n\nu}$ of a user, the asynchronism level $A_n=e^{n\alpha}$
in a slotted strongly asynchronous channel, and the ability to distinguish
$M_n=e^{nR}$ messages per transmission with vanishingly error probability is
investigated in the asymptotic regime as blocklength $n$ goes to infinity. The
receiver must locate and decode, with vanishing error probability in $n$, all
of the transmitted messages. Achievability and converse bounds on the trade-off
among $(R,\alpha,\nu)$ is derived. For cases where $\nu=0$ and $ R=0$,
achievability and converse bounds coincide. A second model for a bursty user
with random access in which the user may access and transmit a message in each
block with probability $e^{-n\beta}$ in then considered. Achievability and
converse bounds on the trade-off between $(R, \alpha, \beta)$ is also
characterized. For cases where $\beta =\alpha$ and $R=0$, the achievability and
converse bounds match.
