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We extend our earlier studies of a multicomponent system accumulating damage due to a series of fatal and nonfatal shocks. The model introduces statistical dependence among the system components by associating individual shock processes with potentially overlapping subsystems made up of groupings of components. We construct an aging and statistical dependence model where damage accumulates multiplicatively with each shock. We derive a representation of the system's joint survival function, and show that the Marshall-Olkin multivariate exponential model can be obtained as a special case of this model. We propose an approach to the simulation of the performance of the system, and provide several illustrative examples. We conclude by identifying possible further extensions of this model.