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This paper investigates the self-sustainability of an overlay Internet of Things (IoT) network that relies on harvesting energy from a downlink cellular network. Using stochastic geometry and queueing theory, we develop a spatiotemporal model to derive the steady state distribution of the number of packets in the buffers and energy levels in the batteries of IoT devices given that the IoT and cellular communications are allocated disjoint spectrum. Particularly, each IoT device is modelled via a two-dimensional discrete-time Markov Chain (DTMC) that jointly tracks the evolution of the data buffer and energy battery. In this context, stochastic geometry is used to derive the energy generation at the batteries and the packet transmission success probability from buffers taking into account the mutual interference from other active IoT devices. To this end, we show the Pareto-Frontiers of the sustainability region, which define the network parameters that ensure stable network operation and finite packet delay. Furthermore, the spatially averaged network performance, in terms of transmission success probability, average queueing delay, and average queue size are investigated. For self-sustainable networks, the results quantify the required buffer size and packet delay, which are crucial for the design of IoT devices and time critical IoT applications.