In this paper we derive a new explicit formula for the Laplace transform of the length of the busy period for the $M|G|1$ queue by a direct martingale method of independent interest. The method is probabilistic, of general character and avoids tedious calculations with complex variables.
Walter A. Rosenkrantz. "Calculation of the Laplace Transform of the Length of the Busy Period for the M|G|1 Queue Via Martingales." Ann. Probab. 11 (3) 817 - 818, August, 1983. https://doi.org/10.1214/aop/1176993531