Modeling a set of probabilistic concurrent processes I'm looking into discrete-time Markov chains (DTMCs) for use in analyzing a probabilistic consensus protocol. One basic thing I haven't been able to figure out is how to model a set of independent processes: consider $N$ processes. These processes will concurrently execute a series of identical instructions labeled $0, 1, 2, 3,$ etc. and all are starting in instruction $0$. When probability is not involved, modeling this is simple: it's a state machine which branches nondeterministically off the start state to $N$ different states, where in each of those $N$ states a different process was the first to execute instruction $0$. What do we do when probability is involved? Do we do the same thing with $N$ states branching from the start state, where the probability of transitioning to each state is $\frac{1}{N}$? As in, it's uniformly random which process was the first one to execute instruction $0$?

Is this like taking the product of the state machines of each process?

I'm using a DTMC here, would I gain anything by moving to a CTMC if I don't care about anything beyond the global order of execution?

Bonus question: assigning probabilities to whichever action is taken first seems like a generalization of the non-probabilistic notion of fairness; if it is, what is the formal definition of this generalized notion of probabilistic fairness?
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 wanted.

Now the transitions of the entire system, and their probabilities, should be easy to derive.  A state of the system is of the form $\langle s_0,s_1,\dots,s_N \rangle$ where $s_0$ is the state of the scheduler and $s_i$ is the state of the $i$th basic process.  Each transition might have the form $\langle s_0,\dots,s_N \rangle \to \langle s'_0,s_1,\dots,s_{i-1},s'_i,s_{i+1},\dots,s_N \rangle$ (where $i$ is a deterministic function of $s_0$, and with probability given by the probability of $s_0 \to s'_0$ in the scheduler multiplied by the probability of $s_i \to s'_i$ in the $i$th basic process).

There are probably other ways to model it as well, maybe better ones for all I know.  This is just one possibility that occurs to me.