We consider the lead time of priority lots in a semiconductor factory. The main cause of delays is waiting for repairs of failed machines, and the failures are mainly caused by power disturbances. This can be modeled as an exogenous time-dependent failure process, and we develop a probabilistic model of the system based on this. Using this model, a convolution algorithm for finding the lead time distribution is described. We describe a method of creating synthetic samples of the lead time from historical failure and repair data collected in the factory. Based on such a sample from the MIT Integrated Circuits Laboratory, we approximate the distribution of lead times by a gamma probability density function shifted by the smallest possible lead time for the product type. The parameters of the gamma distribution are found by using a maximum likelihood estimator. The resulting distribution gives good agreement with the synthetic data for values less than two standard deviations above the mean lead time. Since our procedure only depends on a description of the process and the failure and repair history of the factory, it can also be used to obtain lower bounds on the lead time for new product types.