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The introduction of heterogeneous wireless mesh technologies provides an opportunity for increased network capacity, broader coverage and higher quality of service (QoS). Each wireless device uses different standards, data formats, protocols, and access technologies. However, the diversity and complexity of such technologies pose challenges for traditional control and management systems. This article proposes a heterogeneous metropolitan area network architecture that combines an IEEE 802.11 wireless mesh network (WMN) with a long-term evolution (LTE) network. In addition, a new heterogeneous routing protocol and a routing algorithm based on reinforcement learning called heterogeneous cognitive routing are proposed to select the appropriate transmission technology based on the parameters of each network. The proposed heterogeneous network overcomes the problems of sending packets over long paths, island nodes, and interference in WMNs and increases the total combined network capacity using unlicensed frequency bands instead of buying more LTE license frequency bands. The work is validated through extensive simulations that indicate that the proposed heterogeneous WMN outperforms the LTE and Wi-Fi networks when used individually. The results of the simulation show that the proposed network achieves up to 200% increase in performance compared to Wi-Fi-only or LTE-only networks.