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# Queue Capacity Utilization Allows You to Predict Many Queues Characteristics

The percent of the time that the arriving work will find the server busy, the average number of items in the queue, the average number of items in the system.
#queueing-theory
Zettelkasten, January 6th, 2023.

Knowing queue capacity utilization ($\rho$) allows you to predict the following:

• the percent of the time that the arriving work will find the server busy;
• the average number of items in the queue;
• the average number of items in the system;
• the percent of overall cycle time is queue time;
• the ratio of cycle time to value-added time.

For $M/M/1/\infty$ Queue, you can predict these characteristics:

• Percent Capacity Utilization $= \rho$
• Percent Unblocked Time $= 1 - \rho$
• Number of Items in Queue $= \cfrac{\rho^2}{1-\rho}$
• Numbers of Items in System $= \cfrac{\rho}{1-\rho}$
• Percent Queue Time $= \rho$
• $\cfrac{\text{Cycle Time}}{\text{Value-Added Time}} = \cfrac{1}{1-\rho}$

This property is helpful from a practical perspective, but it's often tough to directly measure capacity utilization in product development processes. Moreover, it's problematic because the ratio of demand and capacity are individually hard to estimate.

## Questions

• What methods can measure queue capacity utilization in product development processes?
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