Consider an alternating renewal system that can be in one of two states: on or off. Initially it is on and it remains on for a time $Z_1$, it then goes off and remains off for a time $Y_1$, it then goes on for a time $Z_2$, then off for a time $Y_2$ ; then on, and so forth. Suppose that the random vectors $(Z_n, Y_n), n > 1$, are i.i.d.. Then $Z_n, n>1$ are i.i.d. and $Y_n, n> 1$ are also i.i.d..
Suppose that the distribution of $Z_n$ is a Geometric distribution and the distribution of $Y_n$ a Poisson distribution. My question is whether it is possible to compute $\lim_{t\rightarrow \infty} P(\text{system is on at time }t)$?
I am tempted to apply Theorem 3.4.4 of Stochastic processes by Sheldon M. Ross, which states that
If $E[Z_n + Y_n] < \infty$ and $Z_n + Y_n$ is nonlattice, then $\lim_{t\rightarrow \infty} P(\text{system is on at time }t) = \frac{E(Z_n)}{E(Z_n)+E(Y_n)}$
But $Z_n + Y_n, n\geq 1$ are nonnegative integer valued random variables, and therefore lattice, which violates the condition of the theorem.
Thanks for your help!