# Background

Let $X_t$ be a stochastic process on the state space {Working, Broken}. Let $U$ be the cumulative sojourn Working during an interval $[0,\tau]$ (the process's *uptime*). It is well-known [1] that if the process has a finite mean time to failure (MTTF) and mean time to repair (MTTR), then $$
\begin{align*}
P\left\{ X_t = \text{Working}\ | \ X_s = \text{Working} \right\} &= p + (1-p) \cdot z(t;s) \\
P\left\{ X_t = \text{Broken}\ | \ X_s = \text{Broken} \right\} &= 1-p + p \cdot z(t;s)
\end{align*}
$$
for some function $z$, where $p = \frac{\text{MTTF}}{\text{MTTF}+\text{MTTR}}$.

Now further suppose that the times to failure have finite variance, and so too for the times to repair. Then in a classic result [2], Takács showed $U$ is asymptotically normal as $ \tau \rightarrow \infty $, and calculated the distribution's mean $\mu_U$ and variance $\sigma^2_U$.

# My situation

I can prove that $U$ is asymptotically normal and calculate $\mu_U$ and $\sigma^2_U$, but with two caveats:

I need to make an assumption about $z$. Thus my result is

**less general**than Takács's.Calculating $\sigma^2_U$ requires calculating $\lim_{\tau\rightarrow\infty} \int_{0}^{\tau} z(t; 0) \ dt$. Thus my result is

**less useful**than Takács's, as solving $z$ for arbitrary processes is an open question.

However, the result is (perhaps?) of interest as an **elementary application** of Billingsley's central limit theorem for dependent variables under strong mixing [3, Theorem 27.4]. (*And yes*: What happened is that I learned about Billingsley's result through Wikipedia when first reviewing literature, found a proof, and *then* I found Takács result. There must be Murphy's Law variant for this.)

If Prof Billingsley was still alive, I'd probably write to him and offer the result as a homework exercise for a new edition of his book. However, failing that option ...

# My question

... what journal might be good for my result?

(I know I can self-publish to arXiv or equivalent with my employer. I am interested in alternatives.)

Many thanks.

# References

[1] Kishor Shridharbhai Trivedi, 2002, *Probability and statistics with reliability,
queuing, and computer science applications*, Second ed., John Wiley
& Sons, New York.

[2] Lajos Takács, 1959, On a sojourn time problem in the theory of stochastic
processes, *Transactions of the American Mathematical Society*, 93(3), 531--540.

[3] Patrick Billingsley, 2008, *Probability and measure*, John Wiley & Sons, New
York.