% m211oha0.tex (new 03) - A Rat Maze II
%   - cf. also m211oh83.tex (A Rat Maze)
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\begin{center}\tcr{\rmH A Rat Maze II} 
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%\LARGE              
\huge

%\begin{description}  
Consider the following \tcr{system of three chambers} connected by \tcg{passages} as shown. 
% At time $t_0$, a rat is placed in one of the chambers (say in chamber 1).
\vspace{5pt}

\noi\begin{minipage}{8cm}
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\begin{minipage}{10cm}
\vspace*{-25pt}
\noi At time \mc{t = 0}, a rat is placed in one of the chambers (say in \tcg{chamber} \mc{1}).
\end{minipage}\vspace{-25pt}

\noi \tcg{Each minute thereafter}, the rat \tcr{is driven out} of its present chamber by some stimulus and is prevented from re-entering immediately.

\noi \tcr{Assume:} the rat \tcr{chooses} the exits of each chamber \tcr{at random}.
%\end{minipage}
%\vspace{-20pt}\vspace{5pt}
\sps

\noi {\tcr{\bf Question:}} What is the \tcg{probability} that the rat is in a certain chamber \tcr{in the long run}?
\sps

\noi \tcr{{\bf Note:}} If \mc{s_i} denotes the \tcg{state} that the rat is in chamber \mc{i}, then we have a \tcr{Markov chain} with 
\vspace{-15pt}\tcr{transition matrix}
\tcb{$$
\exprow{1.2}
A=\left(\begin{array}{ccc}
0&\frac{1}{2}&\frac{1}{3}\\
\frac{1}{2}&0&\frac{2}{3}\\
\frac{1}{2}&\frac{1}{2}&0
\end{array}\right). 
\exprow{1.0}
$$}

\newcommand{\bmat}{\left(\begin{array}}
\newcommand{\emat}{\end{array}\right)}

\begin{description}
\item[\tcr{Qualitative Solution:}] \mc{A} is \tcr{primitive} because
$$
\exprow{1.2}
\tcb{A^2=\bmat{ccc}
0&\frac{1}{2}&\frac{1}{3}\\
\frac{1}{2}&0&\frac{2}{3}\\
\frac{1}{2}&\frac{1}{2}&0
\emat \bmat{ccc}
0&\frac{1}{2}&\frac{1}{3}\\
\frac{1}{2}&0&\frac{2}{3}\\
\frac{1}{2}&\frac{1}{2}&0
\emat = \frac{1}{12}
\bmat{ccc}
5&2&4\\
4&7&2\\
3&3&6
\emat.} 
\exprow{1.0}
$$
Thus, by \tcg{Perron's Theorem} there is a vector \mc{\vec{p} =(p_1,p_2,p_3)^t} such that: 

-- each \mc{p_j} represents the \tcg{probability} that the rat is in 
chamber \mc{j} \tcr{in the long run:}
$$
\tcb{\lim_{n\ra\infty} \vec{v}_n = \lim_{n\ra\infty} A^n\vec{v}_0 
= \vec{p}}
$$
-- \mc{\vec{p}} does \tcr{not} depend on the \tcg{initial distribution} \mc{\vec{v}_0}, i.e.\ on where the rat was originally placed;  

-- \mc{p_j >0}, for \mc{j=1,\ldots,3}, i.e.\ \tcr{no} chamber is \tcg{excluded} in the long run.

\item[\tcr{Quantitative Solution:}] 
Find the \tcg{Perron vector} \mc{\vec{p}}:
$$
\tcb{\vec{p} = \frac{1}{27}(8,10,9)^t
\doteq (.296,.370,.333)^t}
$$
%where \mc{\vec{x} = (6,16,11,11)^t}. 
\tcr{Thus}, \tcg{in the long run:}

\begin{tabular}{ccl}
The rat spends & \mc{29.6\%} & of its time in chamber \mc{1}, \\ 
               & \mc{37.0\%} & of its time in chamber \mc{2}, \\
               & \mc{33.3\%} & of its time in chamber \mc{3}. 
\end{tabular}
\end{description} 
\end{document}                          