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{\bf COURSE 336 \\ 
PERFORMANCE ANALYSIS\\
P.G. HARRISON \\
Coursework 3\\
03/02/2006\\
Assessed \\
due date: see CATE}
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A  small shop has space for one customer only.  If the shop is empty, then  in
the next time-unit, either a 
customer arrives with probability  $\alpha$ or 
the shop remains empty.  If a customer has arrived,
 then the shop is full, and could either stay 
  full in the next time-unit or the  customer 
  could leave the shop  empty with probability $\beta$.
\begin{enumerate}
\item Describe the behaviour of the shop as a Discrete Markov Chain with two states $1,2$ representing the two states of the shop: empty and full.
Define  the one step probability  matrix $\mathbf{Q}$.
\item Write down  the state probability 
at time $0$,  assuming
 that  initially  the Markov Chain is in the state empty.  Calculate $q^{(2)}_{12}$ and 
 $q^{(2)}_{11}$.
\item Write down  the state probability at time $0$,  assuming
 that  initially the Markov Chain is in the state full.  
 Calculate $q^{(2)}_{12}$  and $q^{(2)}_{11}$. 
\item Given the matrix  
$M = \left ( \begin{array}{rl} 1 & \alpha \\   
 1 & -\beta\end{array} 
 \right)$
 show that 
 $\mathbf{Q}M = M \left(\begin{array}{rl} 1 & 0\\ 	
 0& \omega\end{array} \right).$
 where $ \omega = 1 -\alpha -\beta$.
Hence show that, for $n \geq 0$
\[\mathbf{Q}^{n}M = M  \left(\begin{array}{rl} 1 & 0\\ 			            		   
 0 & \omega^n\end{array} \right).\]
\item Verify that the inverse of the matrix $M$ is
 $ \frac{1}{\alpha+ \beta} \left(\begin{array}{rl} \beta & \alpha\\
 1& -1\end{array} \right)$. Then show that:
 \[  \mathbf{Q}^{n}=  \frac{1}{\alpha + \beta}
\left(\begin{array}{rl} \beta+\alpha\omega^n & \alpha(1- \omega^n) \\ 			            		     \beta(1-\omega^n)& \alpha + \beta\omega^n\end{array} \right).\]


\item   If  $-1  < \omega < 1 $ and $\mathbf{Q}^{\infty} $ is the limit of
 $\mathbf{Q}^{n} $ 
 as $n \rightarrow \infty$, show that the rows   of $\mathbf{Q}^{\infty} $
 are the same $p=(p_1,p_2)$ and satisfy $ p= p\mathbf{Q} $.
 
%%as $n \rightarrow \infty$, show that the rows of $\mathbf{Q}^{\infty} $
%% are the same and satisfy $ p= p\mathbf{Q} $, where  $p=(p_1,p_2)$ and 
%%  $p_2= \lim_{n \rightarrow
%% \infty} q^{(n)}_{12}$ and  $p_1= \lim_{n \rightarrow\infty} q^{(n)}_{21}$
 For which values of $\alpha, \beta$ does $\mathbf{Q}^{n} $  not converge
   $n \rightarrow \infty$?
What property does the Markov chain exhibit? What 
is the significance of  case   $\omega=1$?

\end{enumerate}

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