DES - Transient Analysis

Transient Analysis:

Transient analysis is concerned with the computation of the probabilities $P(X(t)=j)$.

Vector of the state probabilities at time $t$

Let’s define the vector of the state probabilities at time $t$ as:

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

where:

$${\Pi }_j(t)=P(X(t)=j),j=1,2,3,\dots$$

Then we have, using the total probability rule:

$${\Pi }_j(t)=\sum _{\genfrac{}{}{0ex}{}{}{i\in \mathcal{X}}}\underset{\genfrac{}{}{0ex}{}{}{p_{i,j}(t)}}{\underbrace{P(X(t)=j\mid X(0)=i)}}\cdot \underset{\genfrac{}{}{0ex}{}{}{{\Pi }_i(0)}}{\underbrace{P(X(0)=i)}}$$

In Matrix Form:

$$\Pi (t)=\Pi (0)\cdot H(t)$$

Then using a property of the matrix $Q$, we can write:

$$\Pi (t)=\Pi (0)\cdot e^{Qt}$$

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Observation:

$$\frac{d}{dt}\Pi (t)=\frac{d}{dt}\left(\Pi (0)\cdot e^{Qt}\right)$$

Hence, $\Pi (t)$ is the solution of the Cauchy problem.

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~Example: