Mathematical Model for the Optimal Utilization Percentile in M/M/1 Systems: A Contribution about Knees in Performance Curves

Abstract

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.