Future mobile networks supporting Internet of Things are expected to provide both high throughput and low latency to user-specific services. One way to overcome this challenge is to adopt network function virtualization and Multi-access Edge Computing (MEC). In this paper, we analyze an end-to-end communications system that consists of both MEC servers and a server at the core network hosting different types of virtual network functions. We develop a queueing model for the performance analysis of the system consisting of both processing and transmission flows. We provide analytical approximations of the performance metrics such as system drop rate and average number of tasks in the system. Simulation results show that our approximations perform quite well. By evaluating the system under different scenarios, we provide insights for the decision making on traffic flow control and its impact on critical performance metrics.