 Solution

For equal rates, the master equation generator is $f$ times the symmetric path-graph matrixIts eigenvectors may be chosen as $(1,1,1)$, $(1,0,-1)$, and $(1,-2,1)$, with eigenvalues $0,-f,-3f$. Resolve the initial empty state asEach nonstationary eigencomponent decays with its own rate, givingThe remaining probabilities are $p_0(t)=1/3+e^{-ft}/2+e^{-3ft}/6$ and $p_A(t)=(1-e^{-3ft})/3$. These satisfy the full equations and sum to one. The limit $p_B(\infty )=1/3$ agrees with the stationary result. At short times $p_B(t)=f^2{t}^2/2+O(t^3)$: two successive jumps, $0\to A\to B$, are necessary, so a term linear in $t$ would signal an incorrectly included direct transition.