Hello I have following example and dont really know how to solve it:
I need to find a Q-matrix for Markov Chain with 4 states , I = {1,2,3,4} with following conditions:
- There is no escape from state 2
- You cannot reach 3 from state 1
- The element in the third row and second column of the jump matrix is 1/9.
- The waiting time in state 4 is exponentially distributed with parameter 8.
I arrived at $Q = \begin{pmatrix} q_{11} & q_{12} & 0 & q_{14}\\ 0 & 0 & 0 & 0\\ q_{31} & q_{32} & q_{33} & q_{34}\\ q_{41} & q_{42} & 0 & -8\\ \end{pmatrix}$
I dont know how else to proceed, calculating the jump matrix is easy
The Jump Matrix would be $\Pi = \begin{pmatrix} 0 & \frac{q_{12}}{-q_{11}} & 0 & \frac{q_{14}}{-q_{11}}\\ 0 & 1 & 0 & 0\\ \frac{q_{31}}{-q_{33}} & \frac{1}{9} & 0 & \frac{q_{34}}{-q_{33}}\\ \frac{q_{41}}{8} & \frac{q_{42}}{8}& 0 & 0\\ \end{pmatrix}$ Where the rows of the jump matrix add up to 1 but I still find there are too many unknowns..?