We study perturbations of Schwarzschild-de Sitter black holes in semi-open systems by using the Heun functions. For the semi-open system, a partially reflective wall is added around the event horizon. Three aspects of this model are investigated, namely the quasinormal mode (QNM) spectra, the greybody factor (GF), and the exceptional point (EP). For the QNM aspect, we identify three distinct behaviors as the frequency-independent reflectivity \(\cal K\) increases. The first-type modes approach the real axis and form long-lived quasi-bound states. The second-type modes move toward but do not reach the real axis and retain a finite decay rate. The third-type modes eventually lie on the imaginary axis becoming purely decaying modes. For the GF aspect, GFs exhibit strong oscillations controlled by the distance between the potential and the reflective wall with a real constant reflectivity. In contrast, a Boltzmann-type reflectivity produces only small corrections. Finally, by promoting \(\cal K\) to a complex parameter, the modified boundary conditions give rise to a second-order EP. Parameterizing the vicinity of such EP, we observe the mode exchange phenomenon, and the deviation of spectra scale with the square root of the deviation of the parameter, as predicted by a Puiseux series expansion.
