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:


1. I need to make an assumption about $z$. Thus my result is **less general** than Takács's.
2. 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](http://en.wikipedia.org/wiki/Central_limit_theorem#CLT_under_weak_dependence) 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.



Publishing an elementary proof of a less-general and less-useful version of a classic result?
[Real Analysis Exchange](http://www.stolaf.edu/analysis/) has an Inroads section that specifically covers the case of "*a clever new proof of an important theorem*".


