Complete the corresponding transition matrix of probabilities, giving your answers correct to three decimal places.
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Calculate the steady state probabilities, correct to three decimal places.
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\nCrashed [[1]]
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\n\\(\\mathbf{A}=\\begin{pmatrix} \\simplify{1-{p1}}&\\var{p1}\\\\ \\var{p2}&\\simplify{1-{p2}} \\end{pmatrix}\\)
\n\\(\\Pi(n)\\) represents the proportions in each state after \\(n\\) time periods have elapsed.
\nThe steady state probabilities are found by evaluating \\(\\lim\\limits_{n \\to \\infty}\\Pi(n)\\)
\nRecall \\(\\Pi(n)=\\Pi(0)A^n\\)
\n\\(\\implies\\lim\\limits_{n \\to \\infty}\\Pi(n)=\\Pi(0).\\lim\\limits_{n \\to \\infty}A^n=\\Pi(0).E.\\lim\\limits_{n \\to \\infty}D^n.E^{-1}\\)
\nwhere \\(D\\) is the diagonal matrix of eigenvalues and \\(E\\) is the matrix of corresponding eigenvectors.
\n\nEigenvalues:
\n\\(\\lambda_1=1\\)
\n\\(\\lambda_1+\\lambda_2=\\simplify{1-{p1}}+\\simplify{1-{p2}}\\)
\n\\(\\implies \\lambda_2=\\simplify{1-{p1}-{p2}}\\)
\n\nEigenvectors:
\n\\(When\\,\\lambda = 1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,v_1=\\begin{pmatrix} 1\\\\1 \\end{pmatrix}\\)
\n\\(When\\,\\lambda = \\simplify{1-{p1}-{p2}}\\)
\n\\(\\begin{pmatrix} \\var{p2}&\\var{p1}\\\\\\var{p2}&\\var{p1} \\end{pmatrix}\\begin{pmatrix} x\\\\y \\end{pmatrix}=\\begin{pmatrix} 0\\\\0 \\end{pmatrix}\\)
\n\\(\\var{p2}x+\\var{p1}y=0\\)
\n\\(y=-\\frac{\\var{p2}}{\\var{p1}}x\\)
\n\\(let\\, x=\\simplify{100*{p1}}\\implies\\,y=-\\simplify{100*{p2}}\\implies\\,v_2=\\begin{pmatrix} \\simplify{100*{p1}}\\\\-\\simplify{100*{p2}} \\end{pmatrix}\\)
\n\\(E=\\begin{pmatrix} 1&\\simplify{100*{p1}}\\\\1&-\\simplify{100*{p2}} \\end{pmatrix}\\)
\n\n\\(\\lim\\limits_{n \\to \\infty}D^n=\\lim\\limits_{n \\to \\infty}\\begin{pmatrix} 1^n&0\\\\0&\\simplify{1-{p1}-{p2}}^n\\end{pmatrix}=\\begin{pmatrix} 1&0\\\\0&0 \\end{pmatrix}\\)
\n\\(\\lim\\limits_{n \\to \\infty}\\Pi(n)=\\Pi(0).E.\\begin{pmatrix} 1&0\\\\0&0 \\end{pmatrix}.E^{-1}=\\Pi(0)\\begin{pmatrix} \\simplify{100*{p2}/(100({p1}+{p2}))}&\\simplify{100*{p1}/(100*({p1}+{p2}))}\\\\\\simplify{100*{p2}/(100*({p1}+{p2}))}&\\simplify{100*{p1}/(100*({p1}+{p2}))}\\end{pmatrix}\\)
\n\\(\\lim\\limits_{n \\to \\infty}\\Pi(n)=\\begin{pmatrix} \\var{steady1}\\\\\\var{steady2} \\end{pmatrix}\\)
\n", "extensions": [], "preamble": {"css": "", "js": ""}, "statement": "A mail server used by a large company to handle its email traffic has a probability of \\(\\var{p1}\\) of going off-line in need of maintenance in any given hour.
\nThe probability of a crashed server being fixed within a one hour gap is \\(\\var{p2}\\).
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