## 5 Numerical Evaluation Results
### Separated Peak Demands
#### 5.2.4 Impact of Random Traffic Bursts at Operators  
The traffic model from Section 5.1.4 consisted of eNB Poisson packet traffic, whereby the eNB Poisson traffic rates at a given operator \(o\) were set to result in traffic bursts at prescribed times, as examined in Sections 5.2.1 through Sections 5.2.3. We now generalize this traffic model to random traffic bursts as follows. The eNBs continue to generate independent Poisson packet traffic. The eNB Poisson packet rates at a given operator \(o,\ o=1,2\), follow an independent two-state (on and off) Markov chain. In the on state, the 30 eNBs at a given operator generate an aggregate Poisson traffic rate of 20 Mbps; while in the off state, there is no packet generation. Both states have exponentially distributed random sojourn times with a mean of 10 s. The load is varied by adjusting the steady-state probability \(p_{\text{on}}\) of being in the on state and each simulation scenario is run for 1000 s.  
Figure 6a shows the mean eNB queue length as a function of the on-state probability \(p_{\text{on}}\), i.e., effectively as a function of the load level. We observe from Figure 6 that the SDN control achieves eNB queue length reductions across the entire stable load range from a small on-state (burst) probability \(p_{\text{on}}\) up to a load level near the stability limit, which would be reached for \(p_{\text{on}}=0.5\). The eNB queue length reduction appears initially modest for small \(p_{\text{on}}\) because the bursts are rare for low \(p_{\text{on}}\), i.e., the behavior is similar to the individual burst scenario considered in Figures 3 and 4 and thus can be cleared relatively quickly, even without SDN control. For increasing \(p_{\text{on}}\), the bursts become more frequent, the eNB queue backlogs increase and flexible SDN control with \(V=1000\) achieves substantial queue reductions compared to operation without SDN control and compared to a less flexible SDN control with \(V=100\). Figure 6b-d show the cumulative distribution function (CDF) of the eNB queue occupancy for three load levels, represented by different \(p_{\text{on}}\). We observe from Figure 6b-d that the CDF curves for SDN control reach the level of one within a much smaller span of eNB queue lengths than the operation without SDN control. For instance, for the medium load level \(p_{\text{on}}=0.35\), the CDF for the SDN control with \(V=100\) reaches one for a queue length of about 90 kB; whereas, operation without SDN control reaches a CDF level of one only for around 760 kB eNB queue length. Thus, the CDF results indicate vastly reduced variability of the eNB queue length with the SDN control as the SDN control reacts to the traffic bursts by actively re-allocating backhaul resources among the \(O=2\) considered operators. We observe from Figure 6d that for the low eNB queue occupancies in the range up to about 200 kB, operation without SDN control achieves higher probabilities of keeping the eNB queue lengths in this low range than SDN control with \(V=100\). This is mainly because the SDN control strives for fairness. If some eNB has a small queue occupancy compared to the other eNBs, the SDN control balances out the eNB queue occupancies via the centrally coordinated backhaul bandwidth allocation. In particular, the CDF curve for SDN control with \(V=100\) indicates that almost all the queue occupancies occur around the 200 to 250 kB range (the larger \(V=1000\) allows for flexible violations of the fairness constraint while sharing the backhaul bandwidth and thus achieves substantially lower queue occupancies). If some services do not want to be subjected to this fairness guided resource allocation and rather want priority service, then the priorities can be implemented through weights for their utilities. We considered only \(O=2\) operators sharing the overall backhaul resource \(Z\) in this section.