Assume that a Seattle bus company has 1500 buses in a city and they reside in of the three areas. Georgetown, Ballard, or SEA. Each hour (60 minutes) the following transitions occur: a) 90% of Georgetown buses stay in the Georgetown area, 1% go Ballard and 9% go to SEA.
b) 90% of Ballard buses stay in the Ballard area, 9% go Georgetown and 1% go to SEA.
c) 90% of SEA buses stay in the SEA area, 1% go Georgetown and 9% go to Ballard.
1) Find the transition matrix that represents this movement
2) Find the eigenvalues and eigenvectors corresponding to this matrix. (you can use matlab)
3) What is the absolute value or modulus of each of the eigenvalues?
4)What happens after 2 days ( 48 hours) if initially there are 800 buses Ballard, 700 at Georgetown and 0 at SEA? (can use matlab)
5) Given enough time, does it matter how many buses start at each location as long as there are 1500 total. Explain
Transition matrix
$$ \mathbf{A} = % \left[ \begin{array}{ccc} G\to G & G\to B & G\to S \\ B\to G & B\to B & B\to S \\ S\to G & S\to B & S\to S \\ \end{array} \right] % = % \left[ \begin{array}{lll} 0.9 & 0.01 & 0.09 \\ 0.09 & 0.9 & 0.01 \\ 0.01 & 0.09 & 0.9 \\ \end{array} \right] $$
Diagonalize transition matrix
Eigenvalues:
$$ \lambda \left( \mathbf{A} \right) = \left\{ 1, \frac{1}{100} \left(85\pm4 i \sqrt{3}\right) \right\} % = \left\{ 1, 0.85 + 0.069282 i, 0.85 - 0.069282 i \right\} % $$ The modulus for each eigenvalue is $$ | \lambda | = \left\{ 1, \frac{\sqrt{7273}}{100}, \frac{\sqrt{7273}}{100} \right\} \approx \left\{ 1, 0.852819, 0.852819 \right\} $$ The eigenvalues are the entries in the diagonal matrix $\Lambda$ $$ \Lambda = \left( \begin{array}{ccc} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & \lambda_{3} \\ \end{array} \right) $$ The eigenvalues are plotted against the unit circle below.
Eigenvectors:
The eigenvectors are the column vectors in $\mathbf{P}$ $$ \mathbf{P} = \left( \begin{array}{ccc} 1 & -\frac{2 \left(-4 i+5 \sqrt{3}\right)}{-19 i+\sqrt{3}} & -\frac{2 \left(4 i+5 \sqrt{3}\right)}{19 i+\sqrt{3}} \\ 1 & -\frac{-11 i-9 \sqrt{3}}{-19 i+\sqrt{3}} & -\frac{11 i-9 \sqrt{3}}{19 i+\sqrt{3}} \\ 1 & 1 & 1 \\ \end{array} \right) % \approx % \left( \begin{array}{ccc} 1 & -0.5-0.866025 i & -0.5+0.866025 i \\ 1 & -0.5+0.866025 i & -0.5-0.866025 i \\ 1 & 1 & 1 \\ \end{array} \right) % $$ The spectral resolution is $$ \mathbf{A} = \mathbf{P}^{-1} \Lambda \, \mathbf{P} $$ Now we can manipulate a diagonal matrix $\Lambda$ instead of the dense matrix $\mathbf{A}$. For example $$ \mathbf{A}^{48} = \mathbf{P}^{-1} \Lambda^{48} \, \mathbf{P} $$
After two days
The discrete system is $$ x_{n+1} = \mathbf{A} x_{n} $$ where $n$ is the number of hours. The initial state is $$ x_{0} = \left( \begin{array}{r} 800 \\ 700 \\ 0 \end{array} \right) $$ Compute $$ x_{48} = \mathbf{A}^{48} x_{0} = \mathbf{P}^{-1} \Lambda^{48} \, \mathbf{P}\, x_{0} = \left( \begin{array}{r} 500 \\ 500 \\ 500 \end{array} \right) $$
Steady state
$$ \lim_{n\to \infty} \mathbf{A}^{n} = \mathbf{P}^{-1} \Lambda^{n} \, \mathbf{P}\ = \frac{1}{3} \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array} \right) $$ Each location ends up with one third of the input number of buses.