In a completely heterogeneous semi-Markovian random walk in a discrete system of L (> 1) states, the Green's function was found in Laplace space (the Laplace transform of a function is defined with, f ¯ ¯ ( s ) = ∫ ∫ 0 ∞ ∞ e − − s t f ( t ) d t ). Here, the system is defined through the jumping time (JT) PDFs: ψ ψ i j ( t ) connecting state i with state j (the jump is from state i). The solution is based on the path representation of the Green's function, calculated when including all the path probability density functions of all lengths: