Video sharing and entertainment websites have rapidly grown in popularity and now constitute some of the most visited websites on the Internet. Despite the active user engagement on these online video-sharing platforms, most of recent research on online media platforms have restricted themselves to networking based social media sites, like Facebook or Twitter. We depart from previous studies in the online media space that have focused exclusively on demand-side user engagement, by modeling the supply-side of the crowd-contributed videos on this platform. The current study is among the first to perform a large-scale empirical study using longitudinal video upload data from a large online video platform. The modeling and subsequent prediction of video uploads is made complicated by the heterogeneity of video types (e.g. popular vs. niche video genres), and the inherent time trend effects associated with media uploads. We identify distinct genre-clusters from our dataset and employ a self-exciting Hawkes point-process model on each of these clusters to fully specify and estimate the video upload process. Additionally, we go beyond prediction to disentangle potential factors that govern user engagement and determine the video upload rates, which improves our analysis with additional explanatory power. Our findings show that using a relatively parsimonious point-process model, we are able to achieve higher model fit, and predict video uploads to the platform with a higher accuracy than competing models. The findings from this study can benefit platform owners in better understanding how their supply-side users engage with their site over time. We also offer a robust method for performing media upload prediction that is likely to be generalizable across media platforms which demonstrate similar temporal and genre-level heterogeneity.