# Limiting distribution of alternating renewal process

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.

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## 1 Answer

There is a remark at the end of Theorem 8.23 in Modeling and analysis of stochastic systems By Vidyadhar G. Kulkarni:

If $Y_n+Z_n$ is periodic with period $d$, then the above result is true if $t$ is an integral multiple of $d$.

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What is a periodic random variable? – Did Apr 22 '11 at 19:57
A lattice process is one whose distribution is concentrated at multiples of a period $d$: $f_X(x)=\sum_n c_n \delta (x-n d)$ or $\sum_n P(X_i=nd) = 1$. – Emre Apr 22 '11 at 20:26

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In other words periodic random variable means nothing. – Did Apr 22 '11 at 21:46
@Didier: I think it has meaning. From Ross's book, "A nonnegative random variable X is said to be lattice if there exists $d\geq 0$ such that $\sum_{n=0}^{\infty} P(X = nd) = 1$. That is, $X$ is lattice if it only takes on integral multiples of some nonnegative number $d$. The largest $d$ having this property is said to be the period of $X$." Based on this, I think a nonnegative random variable is called periodic, if it is lattice. – Ethan Apr 22 '11 at 23:18
Based on this one sees that a random variable being lattice is a standard definition and that lattice random variables are very seldom called periodic. – Did Apr 23 '11 at 21:47
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