## 5 Problem Formulation: Mathematical Model and Algorithm  
Now, we provide a simple analysis of the lowerbound of the required bandwidth on the links adjacent to a server. For simplicity, we first define a few notations. * Let \(S=\{0,1,2,\ldots,s-1\}\) be the set of s DASH Servers. * Let \(E\) be the set of edges in the network. * Let \(R=\{0,1,2,\ldots,r-1\}\) be the set of \(r\) requests. * Let \(L=\{0,1,2,\ldots,l-1\}\) be the set of \(l\) layers. 0 means the base layer and 1 means the enhancement layer 1, and so forth. * Let \(Q\) be the set of requests to the servers \(S\). * Let \(Q_{i}\) be the set of requests to a server \(i\)
* Let \(Q_{e}\) be the set of requests going through an edge \(e\)
* Let \(C_{e}\) be the capacity of a link \(e\)
* Let \(f_{q,k}\) be the required bandwidth of a request \(q\) for the layer \(k\). For example, if a request 0 needs layers only up to 2, then the bandwidth of the layers higher than 2 is 0 such as \(f_{0,3}=0\). * Let \(b_{q,k}\) be the flag having value of 0 or 1. When the layer \(k\) of a request \(q\) is accepted, \(b_{q,k}=1\). Otherwise, \(b_{q,k}=0\). It should be noted that for any \(k\), if \(b_{q,k}=0\), then \(b_{q,k+1}=0\). * Let \(d_{i}\) be the degree of a server \(i\). With this notation, we can formulate the objective of this problem as follows  
\[\max\sum_{q\in Q}\sum_{k\in L}b_{q,k}\] (1) s.t \[\sum_{q\in Q_{e}}\sum_{k\in L}f_{q,k}b_{q,k}\leq C_{e},\quad e\in E \tag{2}\] \[b_{q,k+1}=0,k\in\{k|b_{q,k}=0\} q\in Q,k\in L. \tag{3}\]  
Basically, we want to maximize the number of layers accepted to the network with the link capacity constraint (2) and the SVC layer constraint (3). Thus, for the given set of queries \(Q\), we need to decide which queries to accept upto which layers. Intuitively, the bottleneck links are the links adjacent to the servers. We can easily compute the amount of bandwith needed for a server \(i\). \[B_{i}=\sum_{q\in Q_{i}}\sum_{k\in L}f_{q,k}b_{q,k}. \tag{4}\]  
The worst case is that all the requests in \(Q_{i}\) need to use a specific link of a server \(i\).Then, the required bandwith is \(B_{i}\), which is likely to be higher than the bandwidth of the link. The best case is when the traffic is distributed evenly to all the links of the server. Then, each link has the following amount of traffic. \[Avg(B_{i})=\frac{\sum_{q\in Q_{i}}\sum_{k\in L}f_{q,k}b_{q,k}}{d_{i}}. \tag{5}\]  
Equation (5) can be written in a different way as follows. \[Avg(B_{i})=\frac{\sum_{k\in L}\sum_{q\in Q_{i}}f_{q,k}b_{q,k}}{d_{i}}. \tag{6}\]  
Basically, Equation (5) roughly represents the processing mode of allLayer. For a request \(q\), we accept all the layers of the request \(q\). On the other hand Equation (6) represents the processing mode of layerWise. For a layer \(k\), we accept the layer \(k\) from all the requests. Actually, the total required bandwith can be approximated as \(f\times r\), where \(f\) is the average required bandwidth of a request and \(r\) is the number of requests. This is simply the multiplication of the average bandwith and the number of requests. A lower bound of the required bandwidth to thelinks adjacent to the servers can be computed as follows. We assume that the requests are distributed evenly to each of \(s\) servers. Thus each server needs \(\frac{f\times r}{s}\). Finally, each server has \(d\) links. In a best case, the traffic is evenly distributed to each link. Thus, we have  
\[\text{Average Bandwidth}=\frac{f\times r}{s\times d} \tag{7}\]  
The average bandwidth given in (7) is can be considered as the lower bound of the required bandwidth. In practice, some links may need higher bandwidth and other links may need lower bandwidth. However, this can be a good indicator to determine how many requests can be accepted. The detailed algorithm is described in Algorithm 1 by using the notations in Table. 1.