Question

1. Consider the following stochastic matrices Pa = egin{pmatrix} 1 & 0 & 0 \ 0 & frac{1}{2} & frac{1}{2} \ frac{3}{4} & 0 & frac{1}{4} end{pmatrix}, Pb = egin{pmatrix} frac{2}{3} & 0 & frac{1}{3} \ 0 & frac{1}{3} & frac{2}{3} \ frac{1}{2} & frac{1}{2} & 0 end{pmatrix}, Pc = egin{pmatrix} frac{1}{2} & frac{1}{4} & frac{1}{8} & 0 & frac{1}{8} \ 0 & 1 & 0 & 0 & 0 \ frac{1}{4} & frac{1}{4} & frac{1}{2} & 0 & 0 \ 0 & 0 & frac{1}{2} & 0 & frac{1}{2} \ 0 & 0 & 0 & 0 & 1 end{pmatrix}, Pd = egin{pmatrix} 0 & frac{1}{4} & frac{1}{4} & frac{1}{2} & 0 \ frac{1}{2} & 0 & 0 & 0 & frac{1}{2} \ 0 & 0 & frac{7}{8} & frac{1}{8} & 0 \ 0 & frac{1}{4} & 0 & frac{3}{4} & 0 \ frac{1}{2} & 0 & 0 & 0 & frac{1}{2} end{pmatrix}. (i) Draw state transition diagrams for the DTMCs having Pa, Pb, Pc, and Pd as their one-step transition matrices. (ii) Determine the communicating classes for each of the DTMCs having Pa, Pb, Pc, and Pd as their one-step transition matrices. (iii) Classify the states in the classes as either recurrent or transient.

          1. Consider the following stochastic matrices

Pa = egin{pmatrix} 1 & 0 & 0 \ 0 & frac{1}{2} & frac{1}{2} \ frac{3}{4} & 0 & frac{1}{4} end{pmatrix}, Pb = egin{pmatrix} frac{2}{3} & 0 & frac{1}{3} \ 0 & frac{1}{3} & frac{2}{3} \ frac{1}{2} & frac{1}{2} & 0 end{pmatrix},

Pc = egin{pmatrix} frac{1}{2} & frac{1}{4} & frac{1}{8} & 0 & frac{1}{8} \ 0 & 1 & 0 & 0 & 0 \ frac{1}{4} & frac{1}{4} & frac{1}{2} & 0 & 0 \ 0 & 0 & frac{1}{2} & 0 & frac{1}{2} \ 0 & 0 & 0 & 0 & 1 end{pmatrix}, Pd = egin{pmatrix} 0 & frac{1}{4} & frac{1}{4} & frac{1}{2} & 0 \ frac{1}{2} & 0 & 0 & 0 & frac{1}{2} \ 0 & 0 & frac{7}{8} & frac{1}{8} & 0 \ 0 & frac{1}{4} & 0 & frac{3}{4} & 0 \ frac{1}{2} & 0 & 0 & 0 & frac{1}{2} end{pmatrix}.

(i) Draw state transition diagrams for the DTMCs having Pa, Pb, Pc, and Pd as their one-step transition matrices.
(ii) Determine the communicating classes for each of the DTMCs having Pa, Pb, Pc, and Pd as their one-step transition matrices.
(iii) Classify the states in the classes as either recurrent or transient.

$1. Consider the following stochastic matrices Pa = eginpmatrix 1 0 0 0 frac12 frac12 frac34 0 frac14 endpmatrix, Pb = eginpmatrix frac23 0 frac13 0 frac13 frac23 frac12 frac12 0 endpmatrix, Pc = eginpmatrix frac12 frac14 frac18 0 frac18 0 1 0 0 0 frac14 frac14 frac12 0 0 0 0 frac12 0 frac12 0 0 0 0 1 endpmatrix, Pd = eginpmatrix 0 frac14 frac14 frac12 0 frac12 0 0 0 frac12 0 0 frac78 frac18 0 0 frac14 0 frac34 0 frac12 0 0 0 frac12 endpmatrix. (i) Draw state transition diagrams for the DTMCs having Pa, Pb, Pc, and Pd as their one-step transition matrices. (ii) Determine the communicating classes for each of the DTMCs having Pa, Pb, Pc, and Pd as their one-step transition matrices. (iii) Classify the states in the classes as either recurrent or transient.$

Added by Michael E.

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