In this paper, the trade-off between the number of transmissions (or burstiness) $K_n=e^{nν}$ of a user, the asynchronism level $A_n=e^{nα}$ 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 $(R,α,ν)$ is derived. For cases where $ν=0$ 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 $e^{-nβ}$ in then considered. Achievability and converse bounds on the trade-off between $(R, α, β)$ is also characterized. For cases where $β=α$ and $R=0$, the achievability and converse bounds match.