The application of reservoir computing (RC) is for the first time studied in a class of forecasting tasks in which signals are under random physical perturbations, meaning that the data-baring waveform distortions are versatile, and the process is not repeatable. Tumor movement caused by respiratory motion is such a problem and real-time prediction of tumor motion is required by the clinical radiotherapy. In this work, a true-time delay (TTD) respiration monitor based on photonic RC with adjustable nodes connection is developed specifically for this task. A breathing data set from a total of 76 patients with breathing speeds ranging from 3 to 20 breath per minute (BPM) are studied. A double-sliding window technology is demonstrated to enable the real-time establishment of an individually trained model for each patient and the real-time processing of live-streamed tumor position data. Motion prediction of look-ahead times of 66.6 ms, 166.6 ms and 333 ms are investigated. With a 333 ms look-ahead time, the real-time RC model achieves an average normalized mean square error (NMSE) of 0.0246, an average mean absolute error (MAE) of 0.338 mm, an average therapeutic beam exposure efficiency of 94.14% for an absolute error (AE) < 1mm and 99.89% for AE < 3mm. This study demonstrates that real-time RC is an efficient computing framework for high precision respiratory motion prediction.