There are $N$ nodes, each has response time $t_i$ (here and further I use $i$ as an index).
I want probabilistically choose a node with low response time; if there are several nodes with low response time, alternate between them.
The first idea that came to my mind was to make probability of chosing $i$-th node proportional to $1/t_i^2$. But it is not a solution, because if there are many nodes with high response times, then may overweight few good (with low response times) nodes, so that with a big probability there will be selected a bad node.
Formally (using both non-standard analysis and limit on a filter):
I define $p_i$ as a decreasing in each argument positive-valued continuous function of positive numbers $t_i$ as follows: There are $N$ numbers $t_i$. I denote $p_i$ the (the sought for) (standard) probability of the node (that is $p_i$ is a function of $t_i$ and $\sum p_i=1$).
Let extend the above function $p_i$ of $t_i$ by making $t_i$ to be non-standard (positive) reals in such a way that $M$ of them a "good" (infinitely smaller than the rest) and not infinitely bigger than each other.
Let $N\to +\infty$ by adding more "not good" nodes. Find a function of $p_i$ from $t_i$ such that we can warrant that even when $N\to +\infty$ (with the restriction that we don't change existing nodes and add only bad nodes) we have $\sum_{i\in M} p_i \approx 1$ and each $p_i\not\approx 0$.
Help to find such a function of $p_i$ from $t_i$.
Also, I used both non-standard analysis and limit. Isn't it a poor way to formulate things? How to formulate the problem better (concisely and clearly)?