DES - Theorem of 'Existence, Uniqueness and Consistency of the Limit State Probability Vector'

Theorem: Existence, Uniqueness and Consistency of the Limit State Probability Vector:

Let ($\mathcal{X},Q,{\Pi }_0$) be an irreducible CTHMC with all positive recurrent states. Then, there exists an unique limit probability vector:

$$\Pi \triangleq \left[{\Pi }_1,{\Pi }_2,{\Pi }_3,\dots \right]$$

The vector $\Pi$ is such that:

$${\Pi }_i>0\forall i\in \mathcal{X}$$

and the limit probability vector is also consistent

$$\sum _{\genfrac{}{}{0ex}{}{}{i\in \mathcal{X}}}{\Pi }_i=1$$

Analytic Solutions of $\Pi$

Moreover, $\Pi$ can be computed by solving the following linear system of equations:

$$\{\begin{array}{rl}& \Pi \cdot Q=0\\ & \sum _{\genfrac{}{}{0ex}{}{}{i\in \mathcal{X}}}{\Pi }_i=1\end{array}$$

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Corollary:

The theorem above also holds if ($\mathcal{X},Q,{\Pi }_0$) is an irreducible CTHMC and the number of states is finite

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Observation:

Recall that $\Pi (t)$ satisfies the differential equation:

$$\frac{d}{dt}\Pi (t)=\Pi (t)\cdot Q$$

Then we can say:

$$\begin{array}{rl}& \text{If }\Pi (t)\to \Pi ,\text{with: }\Pi \text{ cosntant}\\ & \text{then: }\frac{d}{dt}\Pi (t)\to 0\\ & \text{hence: }\Pi \cdot Q=0\end{array}$$

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Observation:

$\Pi \cdot Q=0$ is a square system, with $Q$ as singular matrix, so it is non-invertible.

$Q$ is singular because: in the properties of the matrix Q is shown that $Q\cdot \vec{1}=0$. The definition of a singular matrix

So $\Pi \cdot Q=0$ has an infinite number of solutions, in order to select the only one that represent a pmf, we add the constraint of consistency: ${\sum }_{\genfrac{}{}{0ex}{}{}{i\in \mathcal{X}}}{\Pi }_i=1$

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~Example:

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