In queueing theory, Kingman’s formula states that the mean waiting time is given by:
AQT = Average Queue Time.
p = Utilization, expressed as decimal
Ca2 + Cs2 = arrival and process coefficient of variations.
τ = average process time
So what are the practical, manufacturing take-aways?
- The longer the average process time is, the more important is the queue length. For example a process that takes a minute, with a queue of 5, is much better than a process that takes an hour with a queue of 5
- Utilization is king. If utilization is low (~50% or less), arrival variation and process variation will have a small impact. If utilization is high (80% or higher), arrival and process variation will matter much more.
- service, and assembly, extra capacity may not be as expensive compared to it’s manufacturing counterparts. This is because it is easy to move/add employees between functions.
- Since many service processes are longer than manufacturing processes, variation is generally more tolerable in service than in manufacturing. In other words, it is generally much more effective to focus on reducing failure demand than variation when it comes to service organizations.
- Reducing time variation is generally less critical than reducing utilization. Utilization is affected by errors which generate failure demand or rework.
- Arrival variation should not be ignored. It is just as important as process variation. Can your arrival variation be influenced through salesmen, incentives, informing customers or reducing supply chain amplifications?
- The work release behavior of processes upstream of bottleneck is important. The smoother (less lumpy) the orders come in at those work stations, the faster the bottleneck will process the orders.
Additionally, the Theory of Constraint advocates 5 steps of improvement: identify, exploit, subordinate, elevate, and repeat. Kingman’s equations gives these additional insights:
-To identify the constraint (bottleneck), it is easier to examine only the load and compare it with demonstrated capacity. Kingman shows that overload begins at less than 100% utilization, and that sensitivity variations is particularly important at high utilization.
-Exploiting constraints should include variation reduction in addition to other methods such as buffering with Inventory.
-Subordinating other resources should include looking upstream of the constraint to examine arrival variation coming into the bottleneck.
Kingman’s equation ties together Lean, 6 Sigma, TOC, and service systems to show that there’s not a one-size fit all when it comes to adding value to your operations. The key is in knowing when to apply each school of thought.
(Be sure to check out John Bicheno’s “The King of Equations” in the Lean Manufacturing Journal for more information and tips related to Kingman’s equation).