I need some help verifying that my understanding of steady state distribution is indeed correct.
I have a transition diagram (model). With around 100 states and 6 variables. I have used a software called PRISM(Probabilistic Model Checker) to calculate the steady state distribution and it generates 100 lines as follows:
Printing steady-state probabilities in plain text format below:
0:(0,0,0,0,0,0)=0.016145825133811565
1:(0,0,1,0,2,0)=0.016145821765883802
2:(0,0,1,1,0,2)=0.016666677916925207
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.
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99:(1,5,1,5,2,1)=0.0016666652544615058
So the general steady state distribution problem is that for:
$$ v\textbf{P} = v $$
Where v is the unknown vector, and P the probability transition matrix. And we want to find v. So I know how to get v and I understand that v represents the vector such that when multiplied with P we get back v. This result describes the long run probabilities and gives us when the probabilities will steady out. So e.g. if calculating the transient probabilities after 20 steps may be the same as doing after 50 steps because the probabilities steady out.
The thing that is confusing me a little is the parentheses generated by PRISM above. E.g. in the first line it says (0,0,0,0,0,0). Is that my 6 variables in the model i.e. the value of the possible states of the variables? And does the first line represent the probability of my model having all 6 variables in state/value = 0?
Let's call my last two variables x and y.
So if someone asked me what is the probability that in the long run, the values of x and y will be 0. So on the first line (0,0,0,0,0,0) this is the case, but on the second line (0,0,1,0,2,0) it is not the case anymore nor is it the case on any of the other 98 lines. So am I correct to state that the probability of both x and y being 0 is simply 0.0164 or 1.6% ?
You now have a two-state graph (a two-by-two matrix) of probabilities.
Use PRISM to calculate its steady state (officially known as its eigenvector).