|Flow Metrics for a Kanban System over time|
(WiP, TiP, DR, Delivery Bias, Net Flow)
In 1961 John Little published his proof of this general queuing theory equation [litt]:
L is the average number of items in the queue, λ is the average arrival rate, and W the average wait timeSince that time Little's Law has found numerous applications in the study of general flow systems from telecommunications to manufacturing, including in Kanban systems. Because throughput or delivery rate is the more significant attribute for management of such systems (and on average it is approximately equal to arrival rate), it is often expressed as follows:
Delivery Rate = WiP / TiP
The overline indicates the arithmetic mean, WiP is the number of items in the system, and TiP the "time in process" from entering to leaving the system under consideration. See this Glossary for further explanation of the meaning of TiP, versus Lead Time or Cycle Time (and why I don't use Cycle Time!).However when we look at data from actual Kanban systems, where the averages are over relatively short periods (say a week or a month), or where average arrival rate does not equal average delivery rate, it is Little's Inequality that applies, not his Law. Juggling the terms above we can express it in this way:
Delivery Rate – ( WiP / TiP ) ≠ 0 ("Little's Inequality")This is because if the system itself is trending in some way (technically the system is not stationary), or if the scope of the averages is not so wide that every item that entered has left the system - usually both these conditions are true for the periods we wish to analyse - then Little's Law does not apply exactly.
That might seem to imply Little's Law is not useful to us. However the degree to which the law is not true is very relevant and does give us important management information:
Delivery Rate – ( WiP / TiP ) < 0 More work is being taken on than is being delivered
Delivery Rate – ( WiP / TiP ) = 0 The system is balanced
Delivery Rate – ( WiP / TiP ) > 0 More is being delivered than new work being taken on
Firstly the figure shows graphs of the 3 main variables in Little's Law: WiP, TiP and Delivery Rate, The next graph is a plot of Little's Inequality - labelled Delivery Bias, showing whether it is greater or less than zero at any point in time. Note that the formula above is normalised relative to the overall average delivery rate for the whole dataset (AvDR) so that the range (in this case between -1 and +1) is comparable with other datasets. The final graph in the set shows Net Flow, the difference between items completed and started - it is one of the metrics included in Troy's spreadsheet and it again provides a view of how balanced the system is.
TiP – WiP / Delivery Rate ≠ 0
- [litt] Little, J. D. C. "A Proof of the Queuing Formula: L = λW," Operations Research, 9, (3) 383-387. (1961)
- [mage] Magennis, T. "Forecasting and Simulating Software Development Projects: Effective Modeling of Kanban & Scrum Projects using Monte-carlo Simulation", FocusedObjective.com. (2011)
- [vaca] Vacanti, Daniel S. "Actionable Agile Metrics for Predictability: An Introduction". LeanPub. (2015)