The existence of good probabilistic models for the job arrival process and the delay components introduced at the different stages of job processing in a Grid environment is important for the improved understanding of the computing concept envisioned by the Grid. In this study, we present a thorough analysis of the job arrival process in the EGEE infrastructure and 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 modeled 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 queuing and Worker Node?s execution times.