A peak pile-up correction of the output pulses measured by a Ge detector is necessary for a quantitative measurement of a pulsed beam of high energy photons which are produced by Compton backscattering of pulsed laser light with relativistic electrons in a synchrotron storage ring. The peak pileup causes not only a distortion of pulse height distribution measured by a Ge detector but also affects quantitative measurements based on integrating the area under the pulse height spectrum. In order to correct the peak pile-up, a Monte Carte simulation has been applied to calculate the pile-up spectrum by superposing double pulse or triple pulse coincidences with different pulse heights and different time delays. It is assumed that the output pulses are randomly generated in time within one burst of the pulsed laser light. An undistorted pulse height spectrum necessary for the simulation has been obtained from a Low counting rate experiment which has a negligible pile-up. A good agreement between the simulated spectrum and the experimental spectrum using different pile-up rates has been obtained. After fitting the experimental data by use of the simulated spectrum, the measured spectrum call be resolved into single, double and triple pile-up components. This results in a quantitatively corrected peak pile-up at different counting rates.