## 4 System and Problem Definition for Network Aware Server Consolidation
### Traffic Pattern Model  
The traffic pattern in a data center network can be modeled in terms of multiple parameters such as flow arrival rates, flow injection rates, flow sizes, flow completion time and proportion of inter-rack and intra-rack flows [42]. In [25] a novel wireless DCN architecture, based on 60 GHz wireless links between the individual servers of namely, S2S-WiDCN was proposed which drastically reduces the power consumption of the network portion of the data center while sustaining comparable performance. The proposed network aware server consolidation can be adopted for S2S-WiDCN or conventional wired fat-tree data center networks. In the S2S-WiDCN, there are six separate directional antenna arrays in the vertical plane of the server, and another one array on the top of the server. Therefore, seven simultaneous links from a server can co-exist at the same time. We represent the number of possible simultaneous links per server as \(\theta\). Let \(F\) be a vector whose elements are the number of existing flows along each sector determined from the number of flows existing in each server based on their destinations and the routing protocol. Let \(f\) denote the traffic flow rate. It is to be noted, that the flow rate \(f\), has a Gaussian distribution [42; 43]. Therefore, to support 99.86% (one-sided z-distribution) of the flow rates, the required channel throughput should be  
\[\mathbf{r}=f_{h+3\sigma}\mathbf{F}, \tag{3}\]  
where elements of the vector \(\mathbf{r}\), are the required channel throughput in each of the sectors and \(f_{h+3\sigma}\) is the value of the flow rate which is three standard deviations higher than the mean. For the S2S-WiDCN, to accommodate multiple channel access, a single 60 GHz IEEE802.11 ad link is subdivided into \(n_{\text{OFDM}}\) number of separate OFDM channels. Therefore, the bandwidth of each OFDM channel is given by,  
\[B_{\text{W}}=B_{60\text{GHZ}}/n_{\text{OFDM}}, \tag{4}\]  
where \(B_{60\text{GHZ}}\) is the bandwidth of the single physical channel. For a wired network, \(B_{\text{W}}\) would be equal to the bandwidth of the connected wire link. Therefore, to reduce the adverse effect of server consolidation on network performance, the following inequality must be satisfied for all wireless links or sectors from each server in the S2S-WiDCN,  
\[r_{x}<B_{\text{W}}\ \ \forall\ x \tag{5}\]  
where \(r_{x}\) is an element of \(\mathbf{r}\). If the inequality in (5) cannot be satisfied due to high flow rates, consolidation will result in worsening of data center network performance as discussed in the results.  
Moreover, it has been observed from the measurement of a variety of data centers in [43], a large proportion of the server-to-server traffic flows, up to 80%, are intra-rack, meaning between servers in the same rack. Only a small remaining proportion of about 20% is inter-rack, or between servers in different racks. Therefore, to reduce the effective load on the network while consolidation, VMs that communicate more often should be migrated into the same physical server. Hence, in addition to reducing server underutilization, co-location of highly communicating VMs is also a desirable goal as it will reduce both power consumption and network traffic. This way, in our consolidation algorithm, we considered both the inequality of (5) and the proportion of inter and intra-rack traffic to make it network-aware.