#hw1.lyx#

#LyX 1.6.7 created this file. For more info see http://www.lyx.org/
\lyxformat 345
\begin_document
\begin_header
\textclass article
\use_default_options true
\language english
\inputencoding auto
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100

\graphics default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize default
\use_geometry true
\use_amsmath 1
\use_esint 1
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\leftmargin 1in
\topmargin 1in
\rightmargin 1in
\bottommargin 1in
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\author "" 
\author "" 
\end_header

\begin_body

\begin_layout Standard
John Hancock
\end_layout

\begin_layout Standard
CEN 6405
\end_layout

\begin_layout Standard
Homework #1
\end_layout

\begin_layout Standard
May 15th, 2014
\end_layout

\begin_layout Section*
12.2
\end_layout

\begin_layout Standard
Exercise 12.2 gives us the following pmf for the distribution of traffic
 arriving at a network gateway:
\begin_inset Formula \[
f\left(x\right)=\left(1-p\right)^{x-1}p,\, x=1,2,3,\ldots,\infty\]

\end_inset


\end_layout

\begin_layout Standard
First, we are required to compute the mean, variance, standard deviation,
 and coefficient of variation of the burst size:
\end_layout

\begin_layout Standard

\series bold
Mean:
\end_layout

\begin_layout Standard
The general formula for the mean value is
\begin_inset Formula \begin{equation}
\mu=E\left(x\right)=\sum_{i=i}^{n}p_{i}x_{i}\label{eq:1}\end{equation}

\end_inset


\end_layout

\begin_layout Standard
In this case, the 
\begin_inset Formula $p_{i}$
\end_inset

 are 
\begin_inset Formula $\left(1-p\right)^{x-1}p$
\end_inset

 for a given value of 
\begin_inset Formula $x$
\end_inset

 in 
\begin_inset Formula $\left\{ 1,2,3,...,\infty\right\} $
\end_inset

, and the 
\begin_inset Formula $x_{i}$
\end_inset

 are the values 
\begin_inset Formula $\left\{ 1,2,3,\ldots,\infty\right\} $
\end_inset

.
 Therefore the mean value is:
\begin_inset Formula \[
\mu=\sum_{i=1}^{\infty}i\left(1-p\right)^{i-1}p\]

\end_inset


\end_layout

\begin_layout Standard

\series bold
Variance:
\end_layout

\begin_layout Standard
The general formula for variance is:
\begin_inset Formula \begin{equation}
Var\left(x\right)=E\left[\left(x-\mu\right)^{2}\right]=\sum_{i=1}^{n}p_{i}\left(x_{i}-\mu\right)^{2}\label{eq:2}\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Again, 
\begin_inset Formula $p_{i}$
\end_inset

 is 
\begin_inset Formula $\left(1-p\right)^{x-1}p$
\end_inset

 for a given value of 
\begin_inset Formula $x$
\end_inset

 in 
\begin_inset Formula $\left\{ 1,2,3,...,\infty\right\} $
\end_inset

.
 We know the value of 
\begin_inset Formula $\mu$
\end_inset

 from 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:1"

\end_inset

.
 We can therefore rewrite 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:2"

\end_inset

 as :
\begin_inset Formula \begin{equation}
Var\left(x\right)=\sum_{i=1}^{\infty}\left[\left(1-p\right)^{i-1}p\right]\left[i-\left(i\left(1-p\right)^{i-1}p\right)\right]^{2}\label{eq:3}\end{equation}

\end_inset


\end_layout

\begin_layout Standard

\series bold
Standard Deviation:
\end_layout

\begin_layout Standard
The square root of the variance is the standard deviation.
 Therefore the standard deviation is the square root of 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3"

\end_inset

 :
\begin_inset Formula \begin{equation}
\sigma=\sqrt{\sum_{i=1}^{\infty}\left[\left(1-p\right)^{i-1}p\right]\left[i-\left(i\left(1-p\right)^{i-1}p\right)\right]^{2}}\label{eq:4}\end{equation}

\end_inset


\end_layout

\begin_layout Standard

\series bold
Coefficient of variation:
\end_layout

\begin_layout Standard
The coefficient of variation is the standard deviation divided by the mean.
 In this case it will be 
\begin_inset Formula $\mu$
\end_inset

 from equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:1"

\end_inset

 divided by 
\begin_inset Formula $\sigma$
\end_inset

 from 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:4"

\end_inset

:
\begin_inset Formula \[
\frac{\sigma}{\mu}=\frac{\sqrt{\sum_{i=1}^{\infty}\left[\left(1-p\right)^{i-1}p\right]\left[i-\left(i\left(1-p\right)^{i-1}p\right)\right]^{2}}}{\sum_{i=1}^{\infty}i\left(1-p\right)^{i-1}p}\]

\end_inset


\end_layout

\begin_layout Standard
To complete the exercise we must also plot the probability mass function
 (pmf) and cumulative density function (CDF) for 
\begin_inset Formula $p=0.2$
\end_inset


\end_layout

\begin_layout Standard

\series bold
pmf plot
\end_layout

\begin_layout Standard
To plot the pmf we plug 
\begin_inset Formula $0.2$
\end_inset

 into the pmf and draw a graph of it as a function of x.
 Plugging 
\begin_inset Formula $0.2$
\end_inset

 into 
\begin_inset Formula \[
f\left(x\right)=\left(1-p\right)^{x-1}p,\, x=1,2,3,\ldots,\infty\]

\end_inset

 gives us 
\begin_inset Formula \[
f\left(x\right)=\left(1-0.2\right)^{x-1}0.2,\, x=1,2,3,\ldots,\infty\]

\end_inset


\end_layout

\begin_layout Standard
This function is pictured below:
\end_layout

\begin_layout Standard
\begin_inset Graphics
	filename /home/john/Documents/school/summer-2014/performance-modeling/12.2-pmf.png

\end_inset


\end_layout

\begin_layout Standard
The CDF of 
\begin_inset Formula $f\left(x\right)$
\end_inset

 is the function:
\begin_inset Formula \[
F\left(x\right)=\sum_{i=1}^{x}\left[\left(1-0.2\right)^{i-1}0.2\right]\,,x=1,2,3,\ldots,\infty\]

\end_inset


\end_layout

\begin_layout Standard
This is a plot of the CDF:
\end_layout

\begin_layout Standard
\begin_inset Graphics
	filename 12.2-cdf.png

\end_inset


\end_layout

\begin_layout Part*
12.11
\end_layout

\begin_layout Standard
The index of central tendency we would use for the list of disk I/O's given
 in exercise 12.11 is the mean.
 
\end_layout

\end_body
\end_document