Shortage of available licensed spectrum is a major barrier to the development of 5G networks. The deployment of long term evolution (LTE) in unlicensed spectrum (LTE-U) is a promising solution to overcome such a barrier. However, the interaction between LTE-U and Wi-Fi in unlicensed spectrum has not been well understood. In this paper, we use stochastic geometry to develop a framework for a multi-radio access technologies (multi-RAT) heterogeneous network, which consists of an LTE-U tier and a Wi-Fi tier. To reduce the intra-and inter-RAT interference, LTE-U employs an ALOHA-like random access scheme and Wi-Fi performs carrier sensing and energy detection before transmission. We derive the coverage probability and spatial throughput of Wi-Fi and LTE-U networks, and perform the asymptotic analysis when the density of Wi-Fi and LTE-U nodes approach infinity. Based on our analysis, we investigate the effect of network parameters on the coverage probability and spatial throughput of these networks. Furthermore, in order to achieve weighted max-min fairness, we optimize the retention probability of LTE-U nodes to maximize the minimum weighted spatial throughput of Wi-Fi and LTE-U networks.
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