-- 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 ![$[t, t+dt]$](/foswiki/pub/Users/Estate/FailureWithRepairs/latex6a92e142bd44882df3ea39e7a63d161c.png)
for small 
, 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.
, 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 for small , 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.