The invention of the laser was certainly one of the most important technological developments of the 20th century. Countless laser applications in our daily life prove the significance of the process of stimulated emission of photons. Besides photons, however, also phonons can be amplified by stimulated emission. This gives rise to the concept of a phonon laser, which represents the sonic equivalent to an optical laser. In recent years, phonon lasers could even be realized experimentally. While optical lasers have been used and studied for more than five decades, the theory of phonon lasers is much less developed. One of the missing links, for example, is a complete phonon laser linewidth theory. This thesis deals with the dependence of the phonon laser characteristics on various system and operating parameters. In particular, the behavior of the linewidth in the vicinity of an exceptional point is examined. Exceptional points occur when the eigenvalues and the corresponding eigenstates of a system coalesce. As we show on a theoretical level, such a non-Hermitian degeneracy results in nontrivial broadening of the phonon laser linewidth - a quantity that should also be accessible in the experiment.