There are definitions of Absolute, Strong and Weak Fairness available across the internet, but I cannot find a definition of K-Fairness property for critical section algorithms (also algorithms that satisfy a 1-fairness or 3-fairness property). Does this actually exist or is this something that was made up to describe a phenomenon that I am not recognizing?
Definition of K-fairness with respect to concurrency?
concurrency
Perhaps you should look for bounded fairness, the property that something happens within the next $k$ times, where $k$ is the bound.