The unrelenting increase in the mobile users' populations and traffic demand drive cellular network operators to densify their infrastructure. Network densification increases the spatial frequency reuse efficiency while maintaining the signal-to-interference-plus-noise-ratio (SINR) performance, hence, increases the spatial spectral efficiency and improves the overall network performance. However, control signaling in such dense networks consumes considerable bandwidth and limits the densification gain. Radio access network (RAN) virtualization via control plane (C-plane) and user plane (U-plane) splitting has been recently proposed to lighten the control signaling burden and improve the network throughput. In this paper, we present a tractable analytical model for virtualized downlink cellular networks, using tools from stochastic geometry. We then apply the developed modeling framework to obtain design insights for virtualized RANs and quantify associated performance improvement.