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With the development of network function virtualization technology, virtual network functions (VNFs) can be chained to provide different services for users in fifth generation access networks. Under the condition of a large number of various user service requests reach the network in a short time, dynamic placement of VNFs is essential to achieve efficient physical resource allocation, while ensuring network stability. Unlike queue models in current works, we design a dynamic scheduling model of two-stage queue to perceive the status of current queue backlog and schedule the queue dynamically. In particular, we formulate a VNF placement cost minimization problem, taking the network stability into account. Owing to that the problem belongs to an NP-hard problem, we transform it into the minimization of the upper bound of the drift-plus-penalty function by the Lyapunov optimization technique. Then, we decompose the reformulated problem into a service function chain (SFC) scheduling problem and an SFC mapping problem, corresponding to our designed dynamic scheduling model of two-stage queue. For the abovementioned two problems, we present a queue-aware dynamic placement of virtual network functions algorithm. It includes a genetic algorithm-based heuristic SFC scheduling algorithm (first stage) and a genetic algorithm-based heuristic SFC mapping algorithm (second stage). Simulation results demonstrate that our placement algorithm can guarantee less placement cost while significantly enhancing network stability, compared with the existing algorithm.