Summary: For evanescent Markov processes with a single transient communicating class, it is often of interest to examine the stationary probabilities that the process resides in the various transient states, conditional on absorption not having taken place. Such distributions are known as quasistationary distributions. We consider the determination of a family of quasistationary distributions of a general level-independent quasi-birth-and-death process. These distributions are shown to have a form analogous to the quasistationary distributions exhibited by birth-and-death processes. We briefly discuss methods for the computation of these quasistationary distributions.