Pilot-job systems have emerged as a computation paradigm to cope with heterogeneity of production grids, greatly improving fault ratios and latency. Tools like DIANE, WISDOM-II, ToPoS and Condor glideIns are now being widely adopted to conduct large-scale experiments on such platforms. However, a model of pilot-job applications is still lacking, making it difficult to determine submission parameters such as the number of pilots to submit to achieve a given performance level. The variability of production conditions and the heterogeneity of the underlying middleware and infrastructure further complicates this issue. This paper presents a performance model for pilot-job applications running on production grids. Based on a probabilistic modelling, we derive statistics about the number of available pilots along time and the makespan of the application given the number of submitted pilots. Results obtained on a radiotherapy application running on the EGEE production grid show that the model...