CUSUM Statistical Monitoring of M/M/1 Queues and Extensions

<div><p>Many production and service systems can be modeled as queueing systems. Their operational efficiency and performance are often measured using queueing performance metrics (QPMs), such as average cycle time, average waiting length, and throughput rate. These metrics need to be quantitatively evaluated and monitored in real time to continuously improve the system performance. However, QPMs are often highly stochastic, and hence are difficult to monitor using existing methods. In this article, we propose the cumulative sum (CUSUM) schemes to efficiently monitor the performance of typical queueing systems based on different sampling schemes. We use M/M/1 queues to illustrate how to design the CUSUM chart and compare their performance with several alternative methods. We demonstrate that the performance of CUSUM is superior, responding faster to many shift patterns through extensive numerical studies. We also briefly discuss the extensions of CUSUM charts to more general queues, such as M/G/1, G/G/1, or M/M/c queues. We use case studies to demonstrate the applications of our approach. Supplementary materials for this article are available online.</p></div