Little's Inequality

The interesting thing about Little's Law, in spite of its very general applicability, and in certain cases mathematical certainty, there are many cases when it simply does not hold true. In those cases it's not so much Little's Law as Little's Inequality. Could the fact that specific instances of data do not follow Little's Law actually give us useful information about our Kanban and Scrum systems and point to the right kind of interventions to manage and improve their flow? That's what this blog is about.

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=λW 

L is the average number of items in the queue, λ is the average arrival rate, and W the average wait time

Since 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

This looks like actionable data for managing flow in Kanban Systems - a number that shows bias in the system towards (or away from) delivering. Let's look at the set of graphs in the figure above that demonstrate this. The data set is from +Troy Magennis's SimResources website and uses his data, spreadsheet and some of the graphs he provides to assist with his forecasting models[mage]. You can find more about Troy's work at FocusedObjective.com and also download these spreadsheets to explore what your data reveals about your Kanban or Scrum systems.

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.

As expected the graphs show a strong correlation between Net Flow and Little's Inequality for most of the range. Clearly if we're starting more than we're finishing we should expect both the inequality and the Net Flow to be negative. What's interesting is where they don't correspond, and why. Look at weeks 2015-11 and 2015-12. Why in week 11 are we finishing more than starting and yet we still have a negative value for Little's Inequality? The clue is in the TiP for these weeks. In week 11 the average TiP is much lower than in week 12. Perhaps this indicates the items closed that week were smaller in size - or maybe they were "expedited" at the expense of other items in progress. When the items that had been in process longer are closed the following week, Little's Inequality indicates more strongly that balance is being restored.

Little's Inequality as expressed above focuses on Delivery Rate, hence the label Delivery Bias. It could be re-expressed to focus on Time In Process as follows:

TiP – WiP / Delivery Rate ≠ 0     

This metric might be labelled Time in Process Bias. We want TiP values to be as low as possible, but a negative value of this metric is likely to indicate an issue to address, since it would indicate that the TiP of the work in progress is likely to be longer than the items recently delivered.

The time an item stays in the process is also the focus of a new metric from +Daniel Vacanti recently published in his Actionable Agile Metrics [vaca] (see also ActionableAgile.com). He also looks at the degree to which a given system follows an ideal flow through a metric he calls "Flow Debt" (roughly translated as delivering more quickly now at the cost of slower times later). Dan prefers the term Cycle Time to Time In Process and so defines Flow Debt as the difference between the "Approximate Average Cycle Time" (AACT) as observed on a Cumulative Flow Diagram and the "Average Cycle Time" (ACT). Comparing these 2 items gives an idea of whether Flow Debt  is being created or not. Flow Debt is accumulating when AACT>ACT. You can calculate AACT by looking at the time since the cumulative arrivals into the system equalled the current cumulative deliveries. ACT is calculated from the arithmetic mean of the actual times for delivered items in the period. Again the degree to which these quantities do not match indicates the degree to which the system is out of balance.

All these metrics - Little's Inequality (or Delivery Bias), Net Flow and Flow Debt - provide insight into the behaviour of Kanban systems based on the degree to which the system follows Little's Law over the period of study. Further experimentation and experience will show the best ways to use them in concert and the best ways to visualise the flow characteristics of the systems and how to intervene to improve them.

If you have data which you would like to analyse using these metrics do let me know. I'm happy to share spreadsheets and advice with anyone who contacts me. Equally check out Troy Magennis's and Dan Vacanti's web sites referenced above for more tools and insights.

References

• [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)