Weibull distributions curves and Lead time metrics
Analyzing team's performance by math and charts.
If you like gathering information from graphics about teams' productivity this post is for you. Probably, you already use Throughput, Cumulative Flow or even the lead time chart. This post presents lead time and histogram graphics and how they’re related with Weibull distributions curves.
The lead time metric
In software development we call "lead time" the space of time that a demand enters the system and it gets out the system. I'm not going too deep in the "lead time" matter now (maybe I'll dedicate a text about it later). So, the graphic below shows basically how much in days (y axis) a demand (x axis) took to be completed.
The Weibull distributions curves
The Weibull distribution, accordingly with [Wikipedia, 2018], is a continuous probability distribution.
The probability density function of a Weibull random variable is:
Where k > 0 is the shape parameter and λ > 0 is the scale parameter of distribution.
Some interesting analysis here is when shape parameter kappa k > 2 (green line), the distribution is near Gaussian shape and when k = 1 (red line), the shape is like a exponential function. Increasing or decreasing the sahpe parameter kappa will change the curve drastically in certain points.
Histogram representation of lead time
So far, so good! The last step before the analysis itself is to show the histogram representation of that lead time chart (Fig. 1).
Comparing Weibull distributions with histogram for lead times
We can clearly see a Weibull distribution with k < 1 shape in Fig. 3. The space of time between the item which took less time (0 ~5.24 days) and the item which took more time (110 days) is too wide to make any reliable forecast. That is what we call “a long tail”. In other words: longer the tail, less predictability we have. This distribution is observed in development teams with unlimited work-in-progress (WIP), lots of blocking issues, too much wasting in workflow, high level of uncertainty, low maturity software development and so on. This distribution also appears in IT operations and IT services.
As [ANDERSON, 2016] wrote:
"If there is less than an even chance of something being delayed then the tail of the distribution gradually gets pulled to the right. Therefore, long duration tasks have more chance of being delayed and things which are delayed have more chance of being delayed further. This is what produces the long tail to the right."
With k = 1 you still have a long tail on the right hand side and so unpredictability.
Improving flow efficiency, limiting WIP, focusing in removing blocking issues and wastes you can see the shape parameter kappa increasing. The value of k would be between 1 and 2. As we can see in Fig. 4 you still have a tail but it is shorter than k ≤ 1 or a “thin tail”. On this case you can make better predictions.
About 2.0 < k < 4.0, in my opinion, is the best shape to make predictions. There’s a “very thin tail”. But in real scenarios of development [ANDERSON, 2016] “pre-emption does happen and some delay is incurred and also given that a specific item of a defined type may still have size and complexity that varies by two orders of magnitude, it is likely that we will see task durations with a distribution curve that has a shape parameter in the range 1.5 < k < 2.0”.
Rarely we'll have k=5 distribution like showed in Fig. 2 (no tail) in real scenarios of development.
Conclusions
We could see that lead times follows Weibull distributions curves. Of course that if you take a good look only at the lead time chart (Fig. 1) you can tell the variation is too wide, making forecasting difficult. But showing histograms and comparing it with Weibull distributions gives you a quick view of a development team's performance and its process maturity.
References
"Weibull distribution". Wikipedia . Accessed April 2018
Anderson, David. "What We Know About Duration: Workflows". 2016
