## 6 Conclusions  
This article has presented a multi-timescale approach for optimizing the sharing of backhaul resources in the SDN-based layered backhaul (LayBack) network architecture. Through primal dual decomposition and Lyapunov drift techniques we decomposed the traditionally centralized SDN resource management into a distributed management model. The distributed resource management accommodates realistic signaling propagation delays by conducting optimization computations at the higher gateway and SDN orchestrator layers at slower timescales compared to the fast-timescale operation at the eNB radio nodes. The distributed optimization is also highly scalable as only slow timescale optimizations of the sharing of the backhaul resources among multiple operators are performed at the central SDN orchestrator; the finer-grained faster timescale resource allocations to the individual eNB radio nodes and various gateway nodes are performed at the lower layers of the multi-layered multi-timescale optimization. Our numerical evaluations for backhaul example networks have quantified the performance characteristics of the described multi-timescale backhaul resource optimization. We found that the SDN controlled sharing of the backhaul resources among operators can significantly reduce the queue lengths at the radio nodes, e.g., the eNBs that serve the upstream traffic from the wireless end users, compared to an optimization without SDN controlled resource sharing. There are many interesting directions for future research on optimizing the backhaul in wireless networks. This present case study has focused on demonstrating the feasibility of a multi-timescale optimization with a specific example optimization methodology (gradient descent combined with Lyapunov drift-plus-penalty method) in a specific configuration of the LayBack backhaul network architecture. Future research should examine how wireless backhaul network architectures should be dimensioned, e.g., how many layers and how many nodes should be in a given layer for a range of anticipated end-device densities and mobility patterns over the geographic area covered by the wireless backhaul network architecture, so as to best support the optimization processes for resource allocation. With the emergence of multi-access edge computing (MEC) it may become important to widen the scope of resource allocation optimization to cover communication, caching (storage), and computation (e.g., virtual machine compute processing) resources [97; 98; 99]. Moreover, the suitability of the various types of optimization methodologies for the resource allocation in wireless backhaul networks should be broadly studied and compared. The comparison should consider both the optimization performance as well as the practical operational aspects, e.g., simplicity and computation resource usage. Another important future research aspect is robustness and reliability of the wireless backhaul network. Emerging cyber-physical systems, such as medical devices that provide critical diagnostics and continuous therapeutic interventions to humans going about their daily lives [100] as well as networked vehicular systems [101; 102; 103], such as the transportation systems in smart cities, require uninterrupted connectivity with high quality of service levels. Wireless backhaul networks need multiple redundant connectivity paths that provide fail-over functionalities in case of failures [104; 105; 106]. Future backhaul resource optimization needs to account for and route among these multiple connectivity paths (e.g., with SDN support [107; 108]) and optimally allocate resources during normal operation as well as after various failure scenarios. Furthermore, it would be of interest to implement the SDN controlled backhaul resource sharing in SDN testbeds [109; 110; 111] to verify the resource sharing performance characteristics in real operational networks. Conceptualization, A.S., L.F., M.R. and A.S.; Methodology, M.W., N.K., L.F., P.S., A.S.T., M.R., and A.S.; Software, M.W., N.K., L.F., and P.S.