Situation
I have a function $f(t)$ with the following known properties:
- $f(t)$ is defined for $t \ge 0$ (i.e. ignore $t < 0$).
- $t_2 \ge t_1 \implies f(t_2) \ge f(t_1)$ (i.e. the function is monotonically decreasing).
- There exists some $t_\alpha$, $0 < t_\alpha < \infty$, for which $t \ge t_\alpha \implies f(t) = 0$.
- Over the interval $0 \le t \le t_\alpha$, the function satisfies ${K \over 2} f(t) \le t_\alpha - t \le {3 K \over 2} f(t)$ for some known positive constant $K$.
Question
Given the above information, can I say anything else about the behavior of $f$? In particular, if I know the value of $f(t)$ over some period $0 \le t \le t_\beta$ where $t_\beta < t_\alpha$, is it possible to predict $t_\alpha$?
If "yes", is the above still consistent if I remove the criteria that the function is monotonically decreasing, and does that make $t_\alpha$ unpredictable?
Background
(Feel free to skip this)
Ultimately, I am attempting to construct a failure model for a fictional process, such that:
- The time of failure ($t_\alpha$) is predetermined and invariant for any given "instance", but variable across "instances".
- At any given point, I can predict the time of failure to within ±50%.
- I cannot predict the exact time of failure until it actually occurs.
The model does not need to follow any existing physical process.
