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In this paper, we characterize Zigbee networks composed by a set of source nodes which transmit to an access point (AP) either directly or through intermediate relay nodes. In the latter case, both uniform and non-uniform clustering configurations are considered. We evaluate the delay (between transmission and reception of data packets) as a function of the network transmission rate (relative to successful packet transmissions) and tolerable network death level (which will be properly defined). Our results show the existence of a characteristic multidimensional performance surface describing the behavior of a Zigbee network in terms of the three mentioned performance indicators. We heuristically derive a closed-form expression for the network performance surface, by interpolating it through the sum of bidimensional Gaussian surfaces.