  Given a positive-integer-valued vector $u=(u_1, \dots, u_m)$ with
$u_1<\cdots<u_m$. A $u$-parking function of length $m$ is a sequence
$\pi=(\pi_1, \dots, \pi_m)$ of positive integers whose non-decreasing
rearrangement $(\lambda_1, \dots, \lambda_m)$ satisfies $\lambda_i\leq u_i$ for
all $1\leq i\leq m$. We introduce a combinatorial construction termed a parking
function multi-shuffle to generic $u$-parking functions and obtain an explicit
characterization of multiple parking coordinates. As an application, we derive
various asymptotic probabilistic properties of a uniform $u$-parking function
when $u_i=cm+ib$. The asymptotic scenario in the generic situation $c>0$ is in
sharp contrast with that of the special situation $c=0$.
