  We consider a process of noncolliding $q$-exchangeable random walks on
$\mathbb{Z}$ making steps $0$ (straight) and $-1$ (down). A single random walk
is called $q$-exchangeable if under an elementary transposition of the
neighboring steps (down, straight) $\to$ (straight, down) the probability of
the trajectory is multiplied by a parameter $q\in(0,1)$. Our process of $m$
noncolliding $q$-exchangeable random walks is obtained from the independent
$q$-exchangeable walks via the Doob's $h$-transform for a certain nonnegative
eigenfunction $h$ with the eigenvalue less than $1$. The system of $m$ walks
evolves in the presence of an absorbing wall at $0$.
  We show that the trajectory of the noncolliding $q$-exchangeable walks
started from an arbitrary initial configuration forms a determinantal point
process, and express its kernel in a double contour integral form. This kernel
is obtained as a limit from the correlation kernel of $q$-distributed random
lozenge tilings of sawtooth polygons.
  In the limit as $m\to \infty$, $q=e^{-\gamma/m}$ with $\gamma>0$ fixed, and
under a suitable scaling of the initial data, we obtain a limit shape of our
noncolliding walks and also show that their local statistics are governed by
the incomplete beta kernel. The latter is a distinguished translation invariant
ergodic extension of the two-dimensional discrete sine kernel.
