It looks like you're new here. If you want to get involved, click one of these buttons!
The network theory course, available in this book, focuses a lot of attention on stochastic Petri nets that are weakly reversible and have deficiency zero. These are the boring ones: both the rate equation and the master equation have well-behaved stable equilibrium solutions, and any solution approaches one of these equilibria.
Of course, mathematicians are interested in boring situations because we can understand them in detail, and learn to distinguish between the boring situations and the interesting situations.
Personally I believe that the rate equation will be a better starting point for understanding these examples than the master equation. The rate equation is easily simulated in a discrete-time context: at each time there's a certain amount of each species (a floating point number), and at each time step we increment or decrement that amount in a straightforward way. I can make this a lot more precise if anyone wants: first I'll write down the rate equation, which is a differential equation, and then I'll discretize it using the Euler method.
However, it will already be somewhat interesting to study them using the programs you folks already have, as long as we can plot the number of things of each species.
Anyway, here are two examples. I'm writing them out as chemical reaction networks rather than stochastic Petri nets, since it takes less space... but it's the same thing.
Here's one from Feinberg's paper:
A <---> A + A
A + B <---> C <---> B
All the reactions are reversible. This has deficiency 1, and for some choices of the rate constants there are 3 equilibria in the same stoichiometric compatibility class, one of which is unstable. If you don't know the jargon, don't worry: the fun part is that we get an unstable equilibrium.
And here's one that approximately describes the Brusselator, a famous reaction with periodic behavior:
0 <---> A ---> B
A + A + B ---> A + A + A
The 0 means `nothing'. In this example the deficiency is 1. It has a periodic solution and an unstable equilibrium where all the species are present in a positive amount. So, the fun is periodic behavior: the amount of the various species cycle around as time passes.