## 5 Numerical Evaluation Results
### Separated Peak Demands
#### 5.2.5 Impact of Random eNB Traffic Bursts  
To evaluate the multi-layer multi-timescale approach for a large-scale network with independent eNB traffic bursts we modify the LayBack architecture from Section 5.1.1 as follows. We consider \(O=20\) operators, each with two GWs; each GW has five eNBs, for a total of 200 eNBs. The overall backhaul capacity still equals \(Z=20\) Mbps, but the operator backhaul capacity is \(Z_{o}=1\) Mbps. We consider this large network for random eNB Poisson packet traffic bursts generated according to an independent two-state (on and off) Markov chain for each of the 200 eNBs. An eNB generates 0.2 Mbps of Poisson packet traffic in the on state (fixed sojourn time of 10 s) and no traffic in the off state (exponentially distributed random sojourn time with mean 20 s for low load, 15 s for  
Figure 6: Mean and CDF of eNB queue length in kB for \(O=2\) operators with random traffic burst as a function of steady-state probability \(p_{\text{on}}\) of burst state. medium load, and 12 s for high load). The stationary distribution of visits to the on and off states is kept at 0.5 and 0.5. The resulting long-run traffic load for the medium load scenario is 16 Mbps (=200-0.2 Mbps-(0.5-10 s)/(0.5-10 s + 0.5-15 s)), while the long-run traffic loads for the low and high load scenarios are 13.3 Mbps and 18.2 Mbps, respectively. We observe from Figure 7a that for the light load scenario the SDN orchestrated backhaul bitrate allocation increases the probabilities for low eNB queue occupancies below 50 kB only relatively slightly compared to the operation without SDN. In contrast, we observe from Figure 7b vastly increased probabilities for low eNB queue occupancies below 50 kB with the SDN control compared to operation without SDN. More specifically, SDN control keeps the eNB queue lengths below 50 kB with a probability near one, whereas queue lengths below 50 kB occur only with a probability of about 0.4 without SDN control. If five or fewer eNBs at a given operator are in the on (traffic burst of 0.2 Mbps) state, then the aggregate traffic of the ten eNBs at the operator can be accommodated within the operator backhaul capacity of \(Z_{o}=1\) Mbps. If six or more eNBs at a given operator are in the traffic burst state and another operator has less than five eNBs in the traffic burst state, then the SDN control can share the backhaul resource among the operators. For the light traffic scenario (Figure 7a), occurrences of six or more simultaneous eNB traffic bursts at a given operator occur only occasionally; thus, there are relatively few opportunities for SDN control to share backhaul resources. For the medium and high load levels (Figure 7b,c) it becomes increasingly likely that the aggregate load from the ten eNBs at a given operator exceeds the operator backhaul capacity \(Z_{o}\). At the same  
Figure 7: Cumulative distribution function (CDF) of eNB queue length for independent eNB traffic bursts with various load levels for long (10 s) bursts and medium load for short (0.5 s) bursts; fixed parameters: \(O=20\) operators, each with two gateways (each with five eNBs). time, due to the general Poisson process clumping behaviors [95; 96], it is likely that the eNB traffic bursts \"clump\" at a given operator and exceed \(Z_{o}\), while other operators have spare backhaul capacity. Thus, central SDN control of the backhaul capacity allocation can achieve substantial eNB queue length reductions compared to the operation without SDN. Regarding the flexibility parameter \(V\), we observe from Figure 7 that the benefit of the large \(V=1000\) relative to the smaller \(V=100\) increases as the load increases from the light/medium load to the high load. This is mainly because, \(1/V\) is essentially the penalty for using spare bandwidth from other operators. For the light and moderate load levels, there are only relatively rare to moderately frequent occasions of bandwidth sharing; thus, there is no pronounced effect of \(V\).