The length of the busy period for a multi-server queueing system is a key performance measure. It has important and wide applications in optimization problems related to queueing systems. Limited by mathematical methods, the busy period probability distribution function of multi-server finite-buffer queueing systems has not been sufficiently investigated for a long time. Using an importantly quantitative relationship between the sub-full busy period distribution of a multi-server finite-buffer Markovian queue and the full-busy period distribution of its corresponding finite waiting space queue, an elegant recursive formula for finding the Laplace-Stieltjes transform of the full busy period distribution of a multi-server Markovian queue with finite capacity is presented. Meanwhile, to calculate the full-busy period distribution function for any given time, the above Laplace-Stieltjes transform in the $s$-domain can be inverted into the time-domain by using the partial fractional decomposition method of rational functions and the rules for calculating residues. A semi-analytic inversion formula for the full busy period distribution function based on the characteristic roots of transform is also presented. Programming with Mathematica and comparing results with the exiting literature in the numerical examples validate the feasibility and simplicity of the proposed approach. This work provides a more intuitive and easy-to-implement way to determine the full-busy distribution function of a multi-server queueing system with finite buffer space.