Generate Kolmogorov-Chapman equations for Markov processes

Question

I am looking for a way to generate Kolmogorov-Chapman equations for MathCad to solve Markov Chain problem. Problem is to find probability of the system being in one of the states. System has N components. I have a graph with 2^N nodes (states), and 2*N parameter: N a's, which is probability that Nth component will break down and N b's, which is the probability that broken component will become healthy again.

N can be close to 10, which means, that there will be 1024 equations at least, so I'm looking for a way to generate these equations.

Mathcad is not a requirement, any other math engine is OK (even Python if then can be solved with smth like scipy)

Answer 1

Problem is to find probability of the system being in one of the states.

As your stochastic process is a combination of N independent Markov chains, the problem is much easier to approach by solving the equations for each chain separately than solving a system of 2^2*N equations for the whole process.

The transition matrix for an individual chain is:

So, the Kolmogorov-Chapman equations are:

And if you still need the equations for the entire system: