  We investigate simple one-dimensional driven diffusive systems with open
boundaries. We are interested in the average on-site residence time defined as
the time a particle spends on a given site before moving on to the next site.
Using mean-field theory, we obtain an analytical expression for the on-site
residence times. By comparing the analytic predictions with numerics, we
demonstrate that the mean-field significantly underestimates the residence time
due to the neglect of time correlations in the local density of particles. The
temporal correlations are particularly long-lived near the average shock
position, where the density changes abruptly from low to high. By using Domain
wall theory (DWT), we obtain highly accurate estimates of the residence time
for different boundary conditions. We apply our analytical approach to
residence times in a totally asymmetric exclusion process (TASEP), TASEP
coupled to Langmuir kinetics (TASEP + LK), and TASEP coupled to mutually
interactive LK (TASEP + MILK). The high accuracy of our predictions is verified
by comparing these with detailed Monte Carlo simulations.
