Cross-layer analysis has been gaining an increasing attention as a powerful tool to study and assess different quality-of-service (QoS) mechanisms in wireless networks. Regarding the physical and data-link layers, in this paper we provide a cross-layer study for visible light communication (VLC) systems operating under statistical QoS constraints, which are inflicted as limits on the delay violation and buffer overflow probabilities. We assume that the VLC access point (AP) is unaware of the channel conditions, thus the AP sends the data at a fixed rate. Under this assumption, and considering an ON-OFF data source, we employ the maximum average data arrival rate at the AP buffer and the non-asymptotic bounds on buffering delay as the main performance measures. Through numerical results, we illustrate the impacts of different physical and data-link parameters on the system performance.