-- Main.estate - 13 Nov 2012

Consider a point process $N(t)$, which counts in time the number of failures up to time $t$. At each failure the item is instantly repaired and as a result, its current failure rate becomes $c$ times smaller. How does this process evolve in time? -- at one hand, the ``baseline" failure rate $\mu(t)$ will typically increase in time, but on the other hand, multiplication by $c$ each time will pull it down. So, what will be the result?

In other words, if $dN(t)$ is the ``small" increment forward, or the number of failures in $[t, t+dt]$ for small $dt$, and if $\mu(t)$ is some ``baseline" failure rate, then this $dN(t)$ is a Poisson random variable with parameter $c^{N(t)} \mu(t)dt$, that is, essentially, the probability of a failure occurring in this interval is $c^{N(t)} \mu(t)dt$, and the process with increment

$dN(t) - c^{N(t)} \mu(t)dt$

is a martingale.

The model can be very naturally generalized to make $c$ random and/or make it changing in time.

It seems that the applications of this model will be numerous, and its mathematical analysis not too simple.
Contact Us | Section Map | Disclaimer | RSS feed RSS FeedBack to top ^

Valid XHTML and CSS | Built on Foswiki

Page Updated: 13 Nov 2012 by estate. © Victoria University of Wellington, New Zealand, unless otherwise stated