# Minimizing the In-Cloud Bandwidth for On-Demand Reactive and Proactive Streaming Applications
## 3 Optimizing Cloud-Based Streaming Internal Bandwidth: Case of an Unpopular Video
### SASST Performance Evaluation: Case of a Probabilistic Load  
In this case, we suppose that the SN request interarrival follows a random variable. Let us denote the average time slots separating two successive requests by \(y\). By applying a recursive Equation (1) to each SN arrival, we obtain  
\[y =y_{s}+0_{t} \tag{14}\] \[2y =2y_{s}+0_{t}\] (15) \[3y =2y_{s}+1y_{t}\] (16) \[\cdots\] \[ny =(\frac{n}{2}+1)y_{s}+(\frac{n}{2}-1)y_{t}\text{ if $n$ is even}\] (17) \[or\] \[ny =(\frac{n+1}{2})y_{s}+(\frac{n-1}{2})y_{t}\text{ if $n$ is odd}. \tag{18}\]  
Figure 4: SASST average bandwidth variation as a function of \(n\).  
By following the same steps as in Section 3.2 (i.e., summing the \(a_{s}\) of Equations (14)-(17) if \(n\) is even and to (18) if \(n\) is odd), we can determine the total FTS slot number denoted as \(Sy_{FTS}\) for \(n\) being even and odd as follows:  
\[Sy_{FTS}(n)=\begin{cases}y[1+2\sum\limits_{i=2}^{\frac{n}{2}}i+(\frac{n}{2}+1)]= y[\frac{n^{2}}{4}+\frac{n}{2}+1],\\ \hskip 142.26378 pt\text{if $n$ is even}\\ y[1+2\sum\limits_{i=2}^{\frac{n+1}{2}}i]= y[\frac{n^{2}}{4}+n-\frac{1}{4}],\\ \hskip 142.26378 pt\text{if $n$ is odd}.\end{cases} \tag{19}\]  
Hence, we can deduce the expression of the average bandwidth value, denoted as \(B_{y}\), consumed by a storage node as follows:  
\[B_{y}(n)=\begin{cases}\frac{\frac{D}{y}+1}{n}+\frac{n}{4}+\frac{1}{2},&\text{ if $n$ is even}\\ \frac{\frac{D}{y}-\frac{1}{4}}{n}+\frac{n}{4}+1,&\text{if $n$ is odd}.\end{cases} \tag{20}\]  
After Resolving \(\frac{dB_{y}}{2n}=0\), we obtain,  
\[n^{*}=\begin{cases}2\sqrt{\frac{D}{y}+1},&\text{if $n$ is even}\\ 2\sqrt{\frac{D}{y}-\frac{1}{4}},&\text{if $n$ is odd}.\end{cases} \tag{21}\]  
Let us replace \(n^{*}\) in \(B_{y}\); then we obtain the optimal value of \(B_{y}\) denoted \(B_{y}^{*}\),  
\[B_{y}^{*}(n^{*})=\begin{cases}\sqrt{\frac{D}{y}+1}+\frac{1}{2},&\text{if $n^{*}$ is even}\\ \sqrt{\frac{D}{y}-\frac{1}{4}}+1,&\text{if $n^{*}$ is odd}.\end{cases} \tag{22}\]  
Assume that the SN request interarrival is an exponential random variable. We can conclude the probability \(p_{0}\) of having zero arrival per time slot \(s\) as follows:  
\[p_{0}=e^{-\lambda s} \tag{23}\]  
where \(\lambda\) is the Poisson average arrival rate.  
Let \(Y\) be a random variable denoting the number of slots \(k\) between two Poisson successive requests, defined by  
\[P[Y=k]=p_{0}^{k-1}(1-p_{0}) \tag{24}\]  
As \(y\) is the average interarrival slot number between two successive requests, we can then express it as follows using Equations (23) and (24):  
\[y=\sum\limits_{k=1}^{+\infty}kP[Y=k]=\frac{1}{1-e^{-\lambda s}}. \tag{25}\]  
Replacing \(y\) by its expression in Equation (22) gives the following:  
\[B_{y}^{*}(\lambda)=\begin{cases}\sqrt{\frac{D}{1-e^{-\lambda s}}+1}+\frac{1}{ 2},&\text{if $n^{*}$ is even}\\ \sqrt{\frac{D}{1-e^{-\lambda s}}-\frac{1}{4}}+1,&\text{if $n^{*}$ is odd}.\end{cases} \tag{26}\]As \(\lim\limits_{\lambda\rightarrow\infty}y=1\), we can deduce the asymptotic values of \(n^{*}\) and \(B^{*}\) as follows:  
\[\lim\limits_{\lambda\rightarrow\infty}n^{*}(\lambda)=\begin{cases}2\sqrt{D+1},& \text{if $n$ is even}\\ 2\sqrt{D-\frac{1}{4}},&\text{if $n$ is odd}.\end{cases} \tag{27}\]  
\[\lim\limits_{\lambda\rightarrow\infty}B^{*}(\lambda)=\begin{cases}\sqrt{D+1}+ \frac{1}{2},&\text{if $n^{*}$ is even}\\ \sqrt{D-\frac{1}{4}}+1,&\text{if $n^{*}$ is odd}.\end{cases} \tag{28}\]  
This shows that the deterministic high load performances done in Section 3.2 are upper bounds of the probabilistic performances of SASST. Another important result achieved through Equation (28) is that the average bandwidth of SASST is less than the average bandwidth of SST, which was expressed in [11; 23] as \(B^{*}_{SST}=\sqrt{2D}-\frac{1}{2}\) for long videos where \(D>3.5\) when \(n^{*}\) is even and for \(D>4.5\) when \(n^{*}\) is odd. Recall that SST was proven to outperform ST in terms of the average consumed bandwidth in [11; 23].