We deal with finite-buffer queueing systems fed by a Markovian point process. This class includes the queues of type M/G/1/N, 𝑀 𝑋 /G/1/N, PH/G/1/N, MMPP/G/1/N, MAP/G/1/N, and BMAP/G/1/N and is commonly used in the performance evaluation of network traffic buffering processes. Typically, such queueing systems are studied in the stationary regime using matrix-analytic methods connected with M/G/1-type Markov processes. Herein, another method for finding transient and stationary characteristics of these queues is presented. The approach is based on finding a closed-form formula for the Laplace transform of the time-dependent performance measure of interest. The method can be used for finding all basic characteristics like queue size distribution, workload distribution, loss ratio, time to buffer overflow, and so forth. To demonstrate this, several examples for different combinations of arrival processes and characteristics are presented. In addition, the most complex results are illustrated via numerical calculations based on an IP traffic parameterization.