  We deal with a discrete-time two-dimensional quasi-birth-and-death process
(2d-QBD process for short) on $\mathbb{Z}_+^2\times S_0$, where $S_0$ is a
finite set, and give a complete expression for the asymptotic decay function of
the stationary tail probabilities in an arbitrary direction. The 2d-QBD process
is a kind of random walk in the quarter plane with a background process. In our
previous paper (Queueing Systems, vol. 102, pp. 227-267, 2022), we have
obtained the asymptotic decay rate of the stationary tail probabilities in an
arbitrary direction and clarified that if the asymptotic decay rate
$\xi_{\boldsymbol{c}}$, where $\boldsymbol{c}$ is a direction vector in
$\mathbb{N}^2$, is less than a certain value $\theta_{\boldsymbol{c}}^{max}$,
the sequence of the stationary tail probabilities in the direction
$\boldsymbol{c}$ geometrically decays without power terms, asymptotically. In
this paper, we give the function according to which the sequence asymptotically
decays, including the case where
$\xi_{\boldsymbol{c}}=\theta_{\boldsymbol{c}}^{max}$. When
$\xi_{\boldsymbol{c}}=\theta_{\boldsymbol{c}}^{max}$, the function is given by
an exponential function with power term $k^{-\frac{1}{2}}$ except for two
boundary cases, where it is given by just an exponential function without power
terms. This result coincides with the existing result for a random walk in the
quarter plane without background processes, obtained by Malyshev (Siberian
Math. J., vol. 12, ,pp. 109-118, 1973).
