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On/off models are common inputs for a variety of communication network models as well as storage and inventory models. A stationary renewal process alternating periods of activity (active transmission, fluid buildup, etc) with periods of inactivity (silence, no transmission, inputs off, etc) induces a stationary indicator process which indicates if the system is active or not. Heavy tails for the on periods induces a covariance function for the indicator process which decreases slowly at a rate characteristic of long range dependence. This has dramatic consequences for fluid models where fluid flows in at constant rate and there is a constant rate of release.