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 (NFV) and Multi-access Edge Computing (MEC). Besides latency constraints, these services may have strict function chaining requirements. In other words, each service has to be processed by a set of network functions in a specific order. The distribution of network functions over different hosts and more flexible routing caused by service function chaining raise new challenges for end-to-end performance analysis. In this paper, as a first step, 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. To approximate the behavior of the system, we decompose the system into subsystems and analyze them independently. By doing so we are able to provide approximate analytical expressions of the performance metrics such as system drop rate, endto-end delay, and system throughput. Then, we show how to apply the similar method to an extended larger system and derive a stochastic model for systems with arbitrary number of servers at the edge.Simulation results show that our approximation model is accurate for the considered systems. We see in Section VI that the simulation and analytical results coincide. 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.Recently, the study of the performance of networks in Virtual Network Function (VNF)/Software Defined Network (SDN) environment has attracted a lot of attention, [7]-[14]. Ye et al. [7]analyze the end-to-end delay for embedded VNF chains. They consider two types of services that traverse different VNF chains and provide the delay analysis for each different chain. 3 Miao et al. [9] provide an analytical model based on Stochastic Network Caclulus (SNC) to provide upper and lower delay bounds of a VNF chain. In their analysis, they consider both the case of bursty and non-bursty traffic. Along similar lines, Duan [8] analyzes an end-toend delay performance of service function chaining for particular services and given resources. Authors in [10]-[14], apply tools from queueing theory to evaluate the performance of systems in SDN environment. In particular, Jarschel et al. [12] study the OpenFlow architecture, where the switch is modeled as an M/M/1 queue and the controller as a feedback system of the delay-loss type M/M/1/S queue. Similarly Goto et al. [13] analyze a simple OpenFlow-based switch in SDN environment, however, they distinguish traffic from the controller and exogenous traffic. Furthermore, a reasonable amount of works consider the modeling of connected VNF as a sequence of queues where the goal is to ...