Explicitly write down and exponentiate the rate matrix whose state space has size $2^N$.

For example this could describe the dynamics of Facebook membership among a group of $N$ people, where person $i$ has instantaneous rate $\mu_i$ of temporarily rage-quitting Facebook because they realize that Facebook is selling their entire activity stream to the NSA.  But person $i$ is then enticed by the combined influence of the size $N-1$ group to re-join Facebook, which happens at an instantaneous rate proportional to the combined influence, where person $j$ has influence $\lambda_{ji}$ if person $j$ is currently on Facebook.