Evolving networks have always been studied a lot featuring the dynamic properties of real-life networks. Studying the mechanism of the growth and death of a network is of great significance to network modeling. Identical to many models focused on the growing process, in this article, we study the decreasing process thoroughly. A novel evolving network model considering the growing and decreasing process is established based on the queueing system. Focused on the degreasing process, we originally investigate two strategies of vertex deleting that are the brutal strategy and the heritable strategy which characterizes the heritable behavior of “dying” vertices in real networks. On the basis of our model, stochastic properties of the proposed network are analyzed, e.g., the distribution and the expectation of the stationary scale of the network are theoretically obtained. In addition to that, degree distributions with different strategies are demonstrated in simulations, which manifests the power-low distribution. The reliability of the network is also studied by different attacks, sharing the same characteristic with the scale-free network.