# ICN-Based Enhanced Content Delivery for CDN
## 4 Problem Formulation
### Model Description
 \tag{2}\]  
When a content is available at the edge server, it will be routed directly to user. However, on encountering a cache miss on the edge server, the routing occurs in two stages. The first is to route the content from a source that is either a router or the origin server in our proposed design, to the CDN edge server. The second is to route content from the edge server to user. Meanwhile, the content is cached along the path. We also assume that IG is placed near edge servers; thus, the delay from IG to an edge server is ignored. The routing process from the origin server to EG is simply regarded as a one-hop transmission with unlimited bandwidth and a large delay. We evaluate the efficiency of completing a content delivery demand based on QoS, which we measure by its most important factor: the total latency from the source to the destination. In the meantime, each demand must fulfill constraints on link and node capacity. To proceed with further discussion, Table 3 summarizes the notations used in this paper. Given the above definitions and assumptions, we formulate our QoS-based optimal design model as follows:  
\[min\sum_{k\in K}(\sum_{n\in N}\sum_{r\in R}(1-y_{n}^{k})[x_{r}^{k}l_{1}^{k}+(1- x_{r}^{k})l_{1}^{k}])+l_{2}^{k} \tag{3}\]  
\[l_{1}^{k}=\sum_{(i,j)\in E}z_{(i,j)}^{k}d_{(i,j)}\ +\ \alpha^{k},\forall\ k\in K,s^{k}\in R,t^{k}\in N,|s^{k}|\geqslant 1 \tag{4}\]  
\[l_{1}^{k}=\sum_{(i,j)\in E}z_{(i,j)}^{k}d_{(i,j)}\ +\ \alpha^{k},\forall\ k\in K,s^{k}\in V_{s},t^{k}\in N \tag{5}\]  
\[l_{2}^{k}=\sum_{(i,j)\in E}z_{(i,j)}^{k}d_{(i,j)}\,\forall k\in K\,s^{k}\in N,t^{k}\in U \tag{6}\]  
subject to constraints (7)-(15): \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Comment** \\ \hline \(V\) & Set of vertices (network nodes) \\ \(E\) & Set of edges (network links) \\ \(U\) & Set of user nodes \\ \(N\) & Set of edge server nodes \\ \(R\) & Set of router nodes \\ \(V_{s}\) & The single origin server node \\ \(F\) & Set of contents \\ \(K\) & Set of demands \\ \(C(v)\) & Finite cache capacity of node \(v\) \\ \(Size(f)\) & Size of content \(f\) \\ \(f=\xi(k)\) & Mapping of demand \(k\) to content \(f\) \\ \(b_{(i,j)}\) & Link capacity of \((i,j)\) \\ \(d_{(i,j)}\) & Link delay of \((i,j)\) \\ \(\gamma_{(i,j)}^{k}\) & Traffic flow related to demand \(k\) at link \((i,j)\) \\ \(\alpha^{k}\) & Latency overhead related to demand \(k\) \\ \hline \(z_{(i,j)}^{k}\) & 0–1 variable, indicating whether \((i,j)\) is used for routing demand \(k\) \\ \(X(f,r)\) & Boolean function of whether to cache content \(f\) at router \(r\) \\ \(Y(f,n)\) & Boolean function of whether to cache content \(f\) at edge server \(n\) \\ \(x_{r}^{k}\) & 0–1 constraint, indicating availability of demand \(k\) at router \(r\) \\ \(y_{n}^{k}\) & 0–1 constraint, indicating availability of demand \(k\) at edge server \(n\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of the notation used in this paper. \[\sum_{j:(i,j)\in E}z^{k}_{(i,j)}-\sum_{j:(j,j)\in E}z^{k}_{(j,i)}= \lambda^{k}_{i},\forall k\in K,(i,j)\in E, \tag{7}\] \[\lambda^{k}_{i}= \begin{cases}1,&i=s\\ -1,&i=t^{k}\\ 0,&i\
eq s^{k}and\,i\
eq t^{k}\end{cases} \] \[\gamma^{k}_{(i,j)}z^{k}_{(i,j)}=\Big{\{}0,\omega^{k}\Big{\}}, \forall k\in K,\forall(i,j)\in E\] (8) \[\sum_{k\in K}z^{k}_{(i,j)}\,\gamma^{k}_{(i,j)}\leqslant b_{(i,j)},\forall(i,j)\in E\] (9) \[\sum_{k\in K}\left[Y(n,\xi(k))\,+\,y^{k}_{n}\right]\leqslant C(n), \forall n\in N\] (10) \[Y(n,\xi(k))\,+\,y^{k}_{n}\,\leqslant\,1,\forall n\in N,k\in K\] (11) \[\sum_{k\in K}\left[X(r,\xi(k))\,+\,x^{k}_{r}\right]\leqslant C(r),\forall r\in R\] (12) \[X(r,f\xi(k))\,+\,x^{k}_{r}\,\leqslant\,1,\forall r\in R,k\in K\] (13) \[y^{k}_{n}\,=\,\{0,1\},\forall k\in K,n\in N\] (14) \[x^{k}_{r}\,=\,\{0,1\},\forall k\in K,n\in N \tag{15}\]  
The objective function (3) minimizes the total content delivery latency over all content. For each demand \(k\), \(l^{k}_{1}\) in Equation (4) denotes the latency to deliver content from an ICN in-network caching router to the edge server.