The 5G mobile network environment has been studied and developed, and the concept of a vEPC (Virtualized Evolved Packet Core) has been introduced as a framework for Network Functions Virtualization (NFV). Machine-to-Machine (M2M) communications in 5G networks require much faster response than are possible in 4G networks. However, if both the control plane (C-plane) and the data plane (D-plane) functions of the EPC are migrated into a single vEPC server, M2M devices and other user equipments (UEs) share the same resources. To accommodate delay-sensitive M2M sessions in vEPC networks, not only signaling performance on the C-plane but also packet processing performance on the D-plane must be optimized. In this paper, we propose a method for optimizing resource assignment of C-plane and D-plane Virtualized Network Functions (VNFs) in a vEPC server, called the vEPC-ORA method. We distinguish the communications of M2M devices and smartphones and model the vEPC server by using queueing theory. Numerical analysis of optimal resource assignment shows that our proposed method minimizes the blocking rates of M2M sessions and smartphone sessions. We also confirmed that the mean packet processing time is kept within the allowable delay for each communication type, as long as the vEPC server has enough VM resources. Moreover, we study a resource granularity effect on the optimal resource assignment. Numerical analysis under a fixed number of hardware resources of MME and S/P-GW is done for various resource granularities of the vEPC server. The evaluation results of numerical analyses showed that the vEPC-ORA method derives the optimal resource assignment in practical calculation times.