## 5 Numerical Evaluation Results
### Results  
#### 5.2.1 Temporal Spacing of Operator Peak Demands  
Overlapping Peak Demands  
We first verify the correct operation of the SDN-based optimization for a scenario that does not permit inter-operator bitrate sharing, specifically, for a scenario where the peak periods of the upstream bitrate demand at the eNBs of the two operators occur simultaneously, as illustrated in Figure 3a. Both operators experience a jump of the demanded upstream bitrate from 5 Mbps to 20 Mbps at simulation time 10 s; the 20 Mbps peak load persists for 10 s, and then returns to the 5 Mbps base loadlevel. Note that these load levels correspond to the prescribed Poisson process rates, i.e., the actual load levels vary according to the stochastic characteristics of the Poisson packet generation processes around the prescribed bitrates, as is visible through the slight random \"ripples\" of the demand bitrates in Figure 3a. We observe from the curves for the allocated operator rates (\(x_{o}\) with SDN, \(\sum_{g\in\mathcal{G}_{o}}y_{g}\) without SDN) in Figure 3a that both the SDN and no-SDN approaches allocate the maximum operator rate of \(Z_{o}=10\) Mbps to serve the peak load. Since both operators experience the peak load at the same time, sharing among operators would not be sensible. Rather, each operator \(o\) should use its own full upstream bitrate resource \(Z_{o}\) to minimize packet delays. We observe from Figure 3a that the SDN-based optimization meets this intuitive optimization goal and gives essentially the same rate allocations as the no-SDN benchmark. In particular, given the equal demands from the eNBs of both operators, the SDN-based optimization strives to allocate an equal share of half of the total upstream backhaul bitrate of \(Z=20\) Mbps to each operator while serving the peak load. Thus, for the entire simulation time duration, the resource allocation with SDN-based optimization follows the resource allocation without SDN. The no-SDN benchmark solves the optimization up to the subproblem Equation (4), whereby the operator upstream transmission capacity is limited to \(Z_{o}=10\) Mbps with the considered parameter  
Figure 3: Upstream traffic demands and corresponding backhaul bitrate allocations to operators as well as aggregated queue length of eNBs associated with a given operator when peak demand periods of the two operators overlap or are separated (fixed parameter: Mean drift-plus-penalty parameter \(V=1000\)): For overlapping peak demands (**a,b**), both the SDN-based optimization and the benchmark without SDN allocate to each operator its maximum capacity of \(Z_{o}=10\) Mbps to serve the peak demands; there is no sharing among operators. For separated peak demands (**c,d**), the SDN orchestrator dynamically shares the total aggregated backhaul capacity of \(Z=20\) Mbps among the two operators, reducing eNB queue lengths compared to the benchmark without SDN-based resource sharing. settings. Thus, by solving subproblem (4), each operator in the no-SDN benchmark is able to allocate up to \(Z_{o}=10\) Mbps when a demand burst occurs. We note that a conventional static allocation of backhaul bitrate (without any dynamic optimization, not even the intra-operator optimization of the no-SDN benchmark) would allocate \(Z_{o}=10\) Mbps for the entire simulation duration. However, only 5 Mbps out of these 10 Mbps could be used during the time period from 0 to 5 s and from 40 s onwards to the end of the simulation time, thus leading to wasted backhaul bandwidth. We observe from Figure 3b that the queue lengths of the eNBs at both operators linearly increase at a constant rate since both operators experience the same peak load that exceeds their respective available backhaul bitrate \(Z_{o}\). In particular, the queue lengths increase from 0 to a maximum value corresponding to 10 s \(\times(20-10)\) Mbps = 100 Mbit while the peak load is feeding into the eNBs from 10 to 20 s simulation time.