In the Internet of Things (IoT), various battery-powered wireless devices are connected to collect and exchange data, and typical traffic is periodic and heterogeneous. Polling with power management is a very promising technique that can be used for communication among these devices in the IoT. In this paper, we propose a novel and scalable model to study the delay and the power consumption performance for polling schemes with power management under heterogeneous settings (particularly the heterogeneous sleeping interval). In our model, by introducing the concept of virtual polling interval, we successfully convert the considered energy-efficient polling scheme into an equivalent purely-limited vacation system. Thus, we can easily evaluate the mean and variance of the delay and the power consumption by applying existing queueing formulae, without developing a new theoretical model as required in previous works. Extensive simulations show that our analytical results are very accurate for both homogeneous and heterogeneous settings.