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.
