# Queue Capacity Utilization

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Book: Principles Of Product Development Flow Product Queues
• The percentage capacity utilization of a queue allows us to model many other variables.

• Percentage unblocked time: $$1 - \rho$$
• Number of items in queue: $$\rho^2/(1-\rho)$$
• Number of items in system: $$\rho/(1-\rho)$$
• Percentage queue time: $$\rho$$
• Cycle time/Value added time = $$1/(1-\rho)$$
• The percentage capacity utilization isn’t directly measurable in product development queues but can be calculated from the above variables.

• In a M/M/1/∞ queue, if we plot the number of items in the queue vs it’s probability given the capacity utilization is constant, we get an exponentially decaying graph

• Basically, the probability of n items being delayed is $$(1-\rho)\rho^n$$

• Thus the probability of n items being delayed decreased exponentially though the delay cost may not.

• For example, if $$\rho=0.75$$ and the cost of 1 item being delayed is $$X$$
• The expected cost of 1 item being delayed is $$0.25*0.75*X = .1875X$$
• The expected cost of 2 items being delayed is $$0.25*0.75^2*2X = .28125$$ which is 50% larger than the cost of 1 item being delayed.