Here's a quick one. It's related to the branching process from Silverfish's answer.
Run a random walk, starting from height 0, say. At each step, either move up by 2 or move down by 1, with probability 1/2 each.
Count the times at which the current height is below the maximum height so far.
The proportion of such times converges to $\phi$.
Fibonacci numbers and Markov chains
I remember a question in which the Fibonacci numbers occurred. While computing the waiting time for the probability of flipping '1-0-0' the probabilities of the state '1' and the state '1-0' are Fibonacci numbers (divided by some power of 2).
We can simulate this in several ways
Generate random binary numbers of ...
I see two viable options in your case (there is probably many more).
Since you have data, you could use bootstrap to resample a,b,c,x,y instead of assuming that they are independent with their respectives means and standard deviations. This will preserve the properties of your data (covariances as you want, but distributions also).
If you ...