We will discuss continuous-time, multi-type branching model of HIV viral dynamics in the blood stream. We are motivated by observations of viral load in HIV+ patients on anti-retroviral treatment (ART). While on ART for HIV, an infected individual's viral load remains non-zero, though it is very low and undetectable by routine testing. Further, blood tests show occasional viral blips: very short periods of detectable viral load. We hypothesize that this very low viral load can be explained principally by the activation of cells latently infected by HIV before the initiation of treatment. Viral blips then represent large deviations from the mean. Modeling this system as a branching process, we derive equations for the probability generating function. Using a novel numerical approach we extract probability distributions for viral load yielding blip amplitudes consistent with patient data. We then compute distributions on duration of these blips through direct numerical simulation. Finally we discuss the implications of our hypothesis on mechanisms of emerging drug resistance, and model extensions intended to address them.