Nt1310 Unit 1 Assignment 1

\section{Towards Solution}
\label{sec:colocation:solution}
We now show analytically how co-locating applications and data together in a micro-cloud reduces energy consumption. In the table below we list all the notations used in the model.
%\newpage
\begin{center}
\begin{tabular}{l | l}
Notations & Descriptions\\ \hline
M & Total Number of micro clouds\\
A & Total Number of Applications\\
H & Total Number of Hosts\\
$E_{ad}$ & active to dormant energy consumption\\
$E_{da}$ & dormant to active energy consumption\\
$E_{sa}$ & sleepy to active energy consumption\\
$E_{as}$ & active to sleepy energy consumption\\
$m_{ic}$& Machine i in a micro-cloud c\\
$R_e$& read energy consumption rate\\
$W_e$& write energy consumption rate\\
$E_{cpu}$

$N_{e}$ is the energy consumption due to data transfer on the local network from the data storage to the computing nodes. If we are computing on the same node on which the data is stored, there will not be any energy consumption because of data transfer on the network. $R_e$ and $W_e$ represent energy consumption due to reading and writing of data on the local disk.
%\[ |x| = \left\{ \begin{array}{ll} % x & \mbox{if $x \geq 0$};\\ % -x & \mbox{if $x < 0$}.\end{array} \right. \]

\item E\_i : energy consumption of idle machines \\ \begin{equation} E_i = \sum_{i=1}^{H} (P_{i} \eta_{i}) \end{equation} where $P_{i}$ is the idle power consumption of machine i and $\eta_{k}$ is the length of the idle period. \end{itemize}

\begin{itemize} \item E\_ad : total energy consumption when transitioning from active to dormant state \\ \begin{equation} E_{ad} = \sum_{k=1}^{H} (\gamma_{k}) \end{equation} where $\eta_{k}$ is the energy consumption when machine k changes state from active to dormant or sleepy

\] \end{itemize}

Substituting equations 2-9 into equation 1, we get:
%\begin{equation}
\begin{multline} E_{total}=(\sum_{i=1}^{A} \sum_{j=1}^{H} \sum_{k=1}^{M} E_{i,j,k}^{c})+(\sum_{i=1}^{H} (P_{i} \eta_{i}))+(\sum_{k=1}^{H} (\gamma_{k}) )+(\sum_{k=1}^{H} (\delta_{k}))+ \\ (\sum_{k=1}^{H} (\theta_{k}))+(\sum_{k=1}^{H} (\lambda_{k}))+ (\alpha (\frac{s(D)}{B_{i,j}}) + (P_{r} R_e) + (P_{w} W_e))+ \\ (\beta (\frac{s(I)}{B_{i,j}})+(P_{r} R_{e}) + (P_{w} W_{e}))+ \sum_{k=1}^{w} (\phi_{k})x_{k}
\end{multline}
%\end{equation}

\subsection{Analysis} Now let's analyze energy consumption for different situations \begin{itemize} \item Data and Applications co-located on active nodes and heat is being used in those nodes micro-clouds \\

\begin{align}
\begin{split}
E_{total} = {}& E_c + E_i + E_{ad} + E_{da} + E_{sa} + E_{as} + E_{trd} + \\ & E_{tri} +E_{co} \\ & = E_c + E_i + 0 + 0 + 0 + 0 + 0 + 0 +0