  Performance curves of queuing systems can be analyzed by separating them into
three regions: the flat region, the knee region, and the exponential region.
Practical considerations, usually locate the knee region between 70-90% of the
theoretical maximum utilization. However, there is not a clear agreement about
where the boundaries between regions are, and where exactly the utilization
knee is located. An open debate about knees in performance curves was
undertaken at least 20 years ago. This historical debate is mainly divided
between those who claim that a knee in the curve is not a well-defined term in
mathematics, or it is a subjective and not really meaningful concept, and those
who define knees mathematically and consider their relevance and application.
In this paper, we present a mathematical model and analysis for identifying the
three mentioned regions on performance curves for M/M/1 systems; specifically,
we found the knees, or optimal utilization percentiles, at the vertices of the
hyperbolas that relate response time as a function of utilization. Using these
results, we argue that an adaptive and optimal queuing system could be deployed
by keeping load and throughput within the knee region.
