## 2 Background and Related Work
### Network Optimization  
[PERSON] et al. [8] introduced the NUM concept to solve the problem of rate allocation in a network with link capacity constraints. Extensive follow-up studies have analyzed the NUM concept in the contexts of distributed optimization and stochastic network theory [9; 10; 11]. For instance, [PERSON] and [PERSON] [34; 35] analyzed Queue length Maximum Weight (QMW) scheduling, which facilitated the subsequent analysis of throughput optimality conditions and related performance guarantees [36; 37]. However, QMW scheduling does not guarantee minimal delay [38], which has led to investigations of QMW variations that reduce delays in general multi-hop networks [39] or provide better delay guarantees [40; 41]. A common shortcoming of these optimization models is that a centralized optimal scheduler can be impractical, mainly due to scalability problems and signaling delays. Decompositions of NUM models can provide the desired implementation scalability to a certain extent. The existing NUM model decompositions strictly conform to a master-slave architecture, which means that these decompositions do not truly reflect the many intermediate layers that exist between the edge devices and the core network in large-scale wireless networks.  
Furthermore, NUM model decompositions commonly build on the so-called _timescale separation assumption_[42; 43], which states that the session interval is much longer than the convergence time of the greedy resource allocation policy [11]. With the timescale separation assumption, the decomposition neglects the convergence of the local control. Building on the timescale separation assumption, decentralized algorithms for link scheduling based on queue lengths have been proposed in [44; 45; 46].  
Our multi-layer multi-timescale NUM framework contains two innovative aspects. First, rather than having a single central master (or single master layer), we consider four decomposition layers that include the SDN controller at the backhaul, the operators, the network gateways, and the eNodeBs. Second, we consider realistic network signaling latencies for the decomposition of the QMW utility over the LayBack architecture layers. The signaling delays make the timescale separation assumption unrealistic. Generally, there have been two categories of studies that have examined the removal of the timescale separation assumption: (1) studies that use intermediate iterates as decisions and assume continuous underlying flows [47; 48], and (2) studies that use a multi-timescale approach across different layers of the protocol stack [49; 50]. More specifically, the study [47] showed that a \(\beta\)-fairness utility function can be maximized, while guaranteeing system stability, under the assumptions that the number of users per class follows a recurrent Markov Chain. We follow a similar rationale as [47] for the intermediate decisions. Moreover, similarly to the second category of studies, we consider multiple timescales. While the different allocation problems in prior studies had been placed in different layers of the conventional protocol stack, the different allocation problems correspond to different layers of the LayBack architecture in our optimization model.  
The Lyapunov drift-plus-penalty method introduced in [51; 52] has been extensively used in recent years for enforcing constraints in dynamic control. We employ the Lyapunov drift-plus-penalty method to incorporate an economic constraint in the allocation across different operators.