  First-order Marcum $Q$-function is observed in various problem formulations.
However, it is not an easy-to-handle function. For this reason, in this paper,
we first present a semi-linear approximation of the Marcum $Q$-function. Our
proposed approximation is useful because it simplifies, e.g., various integral
calculations including Marcum $Q$-function as well as different operations such
as parameter optimization. Then, as an example of interest, we apply our
proposed approximation approach to the performance analysis of predictor
antenna (PA) systems. Here, the PA system is referred to as a system with two
sets of antennas on the roof of a vehicle. Then, the PA positioned in the front
of the vehicle can be used to improve the channel state estimation for data
transmission of the receive antenna that is aligned behind the PA. Considering
spatial mismatch due to the mobility, we derive closed-form expressions for the
instantaneous and average throughput as well as the throughput-optimized rate
allocation. As we show, our proposed approximation scheme enables us to analyze
PA systems with high accuracy. Moreover, our results show that rate adaptation
can improve the performance of PA systems with different levels of spatial
mismatch.
