One of the main use cases for advanced cellular networks is represented by massive Internet-of-things (MIoT), i.e., an enormous number of IoT devices that transmit data toward the cellular network infrastructure. To make cellular MIoT a reality, data transfer and control procedures specifically designed for the support of IoT are needed. For this reason, 3GPP has introduced the Control Plane Cellular IoT optimization, which foresees a simplified bearer instantiation, with the Mobility Management Entity (MME) handling both control and data traffic. The performance of the MME has therefore become critical, and properly scaling its computational capability can determine the ability of the whole network to tackle MIoT effectively. In particular, considering virtualized networks and the need for an efficient allocation of computing resources, it is paramount to characterize the MME performance as the MIoT traffic load changes. We address this need by presenting compact, closed-form expressions linking the number of IoT sources with the rate at which bearers are requested, and such a rate with the delay incurred by the IoT data. We show that our analysis, supported by testbed experiments and verified through large-scale simulations, represents a valuable tool to make effective scaling decisions in virtualized cellular core networks.