## Video on Demand (VoD) Network/Compute Optimization Formulation

In Figure 1, we show a number of data centers used to serve clients located in a number of end user regions. For concreteness, suppose that the data centers are serving video on demand (VoD) streaming services to client regions.

Suppose we had S data centers, $$D{C_s}$$, and that each can stream a maximum quantity $${Q_s}$$ of video. We have R users regions,$${U_r}$$, that have demands $${D_r}$$ for video that must be satisfied. Each $$D{C_s}$$ has a cost of $$c{d_s}$$ to stream a unit of video $$Q$$. In addition, the communication of video from to across the network is subject to network costs and capacity constraints. Let $${q_{sr}}$$ be the total bandwidth of video produced at $$D{C_s}$$ for $${U_r}$$. The objective is to minimize the total cost (data center and network): $\begin{equation}\sum\limits_s {c{d_s}{q_s}} + \sum\limits_{s,r} {networkCost({q_{sr}})} \end{equation}$ Subject to the network capacity constraints as well as: Data center capacity constraints: $\begin{equation}\sum\limits_r {{q_{sr}}} \le {Q_s}\quad \forall s \end{equation}$ And user demands: $\begin{equation}\sum\limits_s {{q_{sr}}} = {d_r}\quad \forall r \end{equation}$