## 2 Background and Related Work
### Wireless Backhaul Network Optimization  
Judicious usage of the resources in backhaul networks can greatly enhance the wireless services while increasing revenues [53; 54]. Generally, the dynamic sharing of installed transmission resources is a promising strategy for enhancing the performance of wireless backhaul networks [55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65]. Our specific focus is on expanding the scope of resource sharing by exploiting the centralized control that SDN provides while operating within the hierarchical layer structure of the SDN-based backhaul networks. We note that aside from the general enhancement of resource sharing and use, some recent optimization studies have sought to consider specific objectives, such as to minimize energy consumption [66; 67; 68], or to optimize for uploading specific content, e.g., video [69].  
Typically, the different hierarchical layers cover geographic regions of different scopes and operate on different timescales, e.g., fast timescales in small localized regions and slow timescales over wide-area regions. To the best of our knowledge, only a few studies have explicitly considered these heterogenous scopes and timescales. [PERSON] et al. [70] combined an allocation of users to a set of beam vectors in the backhaul of a heterogeneous wireless network on a slow timescale with a corresponding transmission time slot allocation on a fast timescale. [PERSON] et al. [71] examined interactions between slow timescale resource allocation in a pool of baseband units (BBUs) in a cloud radio access network with a fast-timescale beam-forming in remote radio heads. The related recent study [72] has examined the interactions between the slicing of the upper layers of the communication network stack at a slow timescale with the fast-timescale wireless channel dynamics, while the study [73] has considered multiple timescales for optimizing a decentralized SDN control structure. Moreover, in the context of computation task scheduling on virtual machines, a collaborative centralized and distributed control approach has recently been examined in [74], while multiple timescales have been examined in [75]. We also note that two timescales have been considered for minimizing energy costs for data center computations [76; 77] and for smart grid optimization [78].  
Complementary to these prior studies, we present a case study on the optimal dynamic allocation of an abstract backhaul resource (represented by a bitrate) over a total of four layers operating on four different timescales. A preliminary version of parts of this case study has appeared in [1]. This article gives a refined comprehensive presentation of our case study, including the complete set of algorithms for solving the four sub-problems at the considered four layers, whereas only the algorithm for solving one subproblem was worked out in [1]. Moreover, this article gives the full details of the evaluation methodology and expanded results.  
We note that this case study does not seek to examine theoretical convergence guarantees for multi-layer multi-timescale optimization. Initial steps towards such a theoretical analysis have recently been reported in [79]. As a complement to and a motivation for detailed theoretical analyses, this present case study seeks to demonstrate the feasibility of the multi-layer optimization with multiple timescales and to showcase performance gains for wireless backhaul.