Suppose I have a length of slab $\Delta z,$ consisting of two types of material. The process $Y(z)$ controls the type of material where the state space is $S = (y_1,y_2).$
In this small interval of slab the process switches to the other material with probability proportional to $\Delta z,$ say $\lambda\Delta z$ and remains the same with probability $1-\lambda\Delta z,$ where $\lambda$ is a positive constant so $Y(z)$ is homogeneous and symmetric in the two materials. The semigroup has the form
$$P_{\Delta z} = \begin{bmatrix} 1-\lambda\Delta z & \lambda\Delta z \\ \lambda\Delta z & 1 - \lambda\Delta z \end{bmatrix}$$ with infinitesimal generator
$$\mathcal{L} = \lambda\begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}.$$ The trajectories of this process are constructed via first introducing the increasing sequence
$$0 < Z_0 < Z_1 < Z_2 < ... < Z_n < ...$$
of successive random transition points, where the process switches material. In this model the layer sizes are
$$Z_1,Z_2 - Z_1,...,Z_n - Z_{n-1},...$$
and form a sequence of independent random variables with the common exponential distribution with parameter $\lambda$
$$\mathbb{P}[Z_n-Z_{n-1}\leq z] = 1 - e^{-\lambda z}$$
for $n\geq 1.$
I am really struggling how to see how this sequence has an exponential distribution. Could someone explain this to me or provide a reference?