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Threshold switching, i.e., the electrical transition from high to low resistivity in amorphous semiconductors, plays a major role in the program/erase processes of phase-change memory (PCM) devices. Understanding and designing materials and cell structures for optimum memory performance require that accurate models for threshold switching be developed, in both the steady-state and the transient regime. This work presents an experimental and modeling study of threshold switching in amorphous chalcogenide materials for PCM applications. The delay time for the onset of threshold switching from the application of an electrical pulse is explained based on 1/f fluctuations of the subthreshold current. A Monte Carlo model is developed for calculating the statistical distribution of delay times. The model can account for 1) the voltage dependence of the average delay time and 2) the statistical spread of the delay time for both rectangular and triangular voltage waveforms. The delay model is used for a reliability assessment of read disturb in PCM devices.