-- Main.estate - 13 Nov 2012
Consider a point process
, which counts in time the number of failures up to time
. At each failure the item is instantly repaired and as a result, its current failure rate becomes
times smaller. How does this process evolve in time? -- at one hand, the ``baseline" failure rate
will typically increase in time, but on the other hand, multiplication by
each time will pull it down. So, what will be the result?
In other words, if
is the ``small" increment forward, or the number of failures in
, and if
is some ``baseline" failure rate, then this
is a Poisson random variable with parameter
, that is, essentially, the probability of a failure occurring in this interval is
, and the process with increment
is a martingale.
The model can be very naturally generalized to make
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.