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One approach is to model it as a composition of $N+1$ processes: a scheduler process, plus your $N$ basic processes. At each time step, the scheduler process selects which basic process will execute next. Your scheduler could be randomized: for instance, at each time step, it could choose which basic process to execute next uniformly at random, if you ...

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If you define the $\mu$-calculus to not allow free variables, you would have the problem that the components of a well-formed formula are not again well-formed formulas. Consider a well-formed formula "$\mu Z . \varphi$", such that $Z$ occurs freely in $\varphi$ and $\varphi$ has no other free variables. If you were to exclude formulas with free variables ...

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The answer to your question very much depends on the values of the probabilities $a_{ij}$. If all are non-zero, then the DTMC forms one single strongly connected component and the probability of reaching $x=3$ is therefore $1$ (it is $1$ for any state in such a case in fact). This result comes from the Long-run theorem, which says that any finite DTMC ...

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