The existence of good probabilistic models for the job arrival process and the delay components introduced at different stages of job processing in a Grid environment is important for the improved understanding of the Grid computing concept. In this study, we present a thorough analysis of the job arrival process in the EGEE infrastructure and of the time durations a job spends at different states in the EGEE environment. We define four delay components of the total job delay and model each component separately. We observe that the job inter-arrival times at the Grid level can be adequately modelled by a rounded exponential distribution, while the total job delay (from the time it is generated until the time it completes execution) is dominated by the computing element's register and queuing times and the worker node's execution times. Further, we evaluate the efficiency of the EGEE environment by comparing the job total delay performance with that of a hypothetical ideal sup...