Introduction to Monte-carlo Analysis for Software Development
2011
Introduction to
Monte-carlo
Analysis for
Software
Development
Forecasting and managing software
development project risks & uncertainty
Monte-carlo analysis is the tool of choice for managing risk in many fields
where risk is an inherent part of doing business. This paper examines how
to use monte-carlo techniques to understand and leverage risk in Software
Development projects and teams.
Troy Magennis
Troy Magennis – Focused Objective Focused Objective (FocusedObjective.com) 1
Page
6/1/2011

Introduction to Monte-carlo Analysis for Software Development
Introduction This paper introduces a technique for
For software development, it is often answering these questions given the risks
necessary to estimate a project upfront in involved in software development and
order to get project approval, obtain budget delivery. Monte-carlo analysis is a proven
and hire the correct team size and skill-mix. technique for determining the likelihood of
This is often at odds with the Agile an outcome in the face of many difficult to
development methodology where full measure input criteria. Monte-carlo analysis
upfront design and specification is avoided, doesn’t completely eliminate any risk, but it
and delivery happens in small iterations does give a much higher degree of
until a backlog is completed. The desire to satisfactory answer than the plain guesses
work iteration to iteration and choose a and gut feel that is employed today (as to
finite level of work each cycle is compelling, release date) in many software projects.
and it does un-deniably bring value to
What is Monte-carlo analysis?
production earlier than a pure waterfall
Monte-carlo analysis is a mathematical
approach. However, the fact still remains
technique that finds the likely patterns in an
that in order to provide any value to an
equations result given random input values
organization, a finite minimum level of
that are constrained between likely real-
functionality (work) needs to be delivered
world values for those inputs. In place of an
by a preferred date, within a budget
equation, for most purposes a spreadsheet
constraint; very few companies will sign off
of software model of the real-world process
on a project that has no target date, and an
is built, and likely (but random) inputs are
open budget. Often delays incur high cost;
fed into these models many thousands of
not just development costs, but also as
times to find a pattern in the results.
competitors launch new feature first, or take
an increasing market share. Even with Agile For example, if you know that there are
teams it is important for any development one-hundred software product stories
manager or organization to be ready to (features) to develop, and that from history
answer the following questions on an (or educated estimate), you know that the
ongoing basis – shortest time it would take each story is one
day, and the longest is three days then a
1. How much will this product cost to
Monte-carlo analysis would simulate in
develop and deliver?
software completing these one-hundred
2. What is the likelihood of releasing
stories with a random work time of between
by date x?
one and three days; and it would do this
3. What resources do you need to hit
thousands of times. The result would be a
date x (money equals people, so the
histogram of the total time for each
question is often how much more
simulated project. This would be similar to
money do you need to hit date x)?
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Introduction to Monte-carlo Analysis for Software Development
if you had the benefit of actually doing the (or anyone put in this position) over-
project one thousand times, but the estimate. They add a little bit more to cover
computer does this quicker. For a model the unknowns – often doubling each
this simple, the answer can be computer by estimate. Worst still, knowing that estimates
simple averaging without employing are traditionally under-estimated, each time
Monte-carlo analysis. But as the model for they are presented, the next level of
developing and delivering software starts to management mentally or in power-point
follow a more real-world scenario, it presentation, double what they see/hear.
quickly gets too complex for simple This leads to projects not being funded
arithmetic. Defects, added scope, because of the excessive investment need
environment downtime and other blocking for even the smallest of features. On the
events, staff availability are just a few of the other end of accuracy, all too often, other
normal day to day events that cause a staff aren’t in this estimate loop, QA for
cascading impact on software delivery, and testing, DevOps for release management,
Monte-carlo analysis is the right tool to graphic designers for the artistic flair, often
manage estimation given the un-predictable don’t get the benefit of adding their input to
nature of these events (but likely following the estimate equation leaving the estimates
a pattern). (even given the contingency fudge factor)
under-estimated for high risk features.
The problem with traditional
estimation The organization as a whole still has the
Developer estimates for software stories problem of needing to make a decision on
often turn out to be in-accurate causing whether the cost involved in a project will
more erosion of trust in organizations than give the return on investment needed to
any other aspect of the business to proceed. For that decision, a delivery date
technology teams’ relationship. When a and the cost (staff, equipment, software
new project is explored, developers are licenses, etc.) of development and delivery
given vague single sentence descriptions of is needed. Given no other option, the
a vision an analyst or business owner has in developer estimates have to be taken as an
their head, and asked to give an estimate. input, and therefore delivery date and
These estimates are totaled, and that total budget are fixed. Through the use of Monte-
divided by a utilization rate for developers, carlo simulation, and the ongoing tuning of
and turned into a number of weeks. From historical patterns of events within an
that time forward the date is fixed, and organization, it is possible to improve the
often the budget. estimates without causing more work by
the developers in estimating, or requiring
Given the vague inputs and knowledge more detailed specification up-front.
they will be held to this estimate, developer
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Introduction to Monte-carlo Analysis for Software Development
Modeling software projects average. It’s common for small
This paper looks at two common Agile stories to be more accurate than
methodologies and presents Monte-carlo massive stories with lots of
models for each. Scrum is a commonly unknown risks, so these adjustments
employed agile development process, and are entered for each estimate size
Kanban is an emerging methodology that level. These can be obtained by
shows great promise in predictable software analyzing actual versus estimate
delivery. data of already completed stories, or
if no data exists initially guessed
Scrum Modeling with Excel
3. Defect rate (expressed as 1 defect for
Scrum delivers value through fixed time
x points of y size)
iterations. Teams choose a set of
4. Added scope rate (expressed as 1
functionality (stories) to deliver each
story for every x points of the
iteration time-box, measured in a points
medium story size for the project, 5
system. For this example we will use
in this example)
Microsoft Excel. The first step to Monte-
5. Start date
carlo simulation is to build a model of a
6. Days per iteration (work days, for
scrum process using various excel formula’s
example 10 for a two week iteration
that cascade into a final amount of story
cycle)
points for each simulation row. From the
7. Number of story points per iteration
story points, the number of iterations, and
targets (team velocity, pick a will
therefore a date can be determined.
always be better than lower velocity,
The inputs required for this model are – a stretch goal velocity for the upper
bound, and the velocity falling
1. Number of stories for each “size” between these two limits as a
story (in this example, stories were starting point)
estimates were limited to one of 1, 2,
3, 5, 8, 13, 20, 40 units by the To capture this data, an input worksheet
developers. Also in this example, can be built in Excel, similar to that shown
some stories were missing estimates, in Figure 1.
so they were spread according to the
median story size of existing story
estimates)
2. The lower bound, average and high
boundary adjustments to apply
against each estimate size. Random
numbers will fall within these
boundaries, weighted towards the
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Introduction to Monte-carlo Analysis for Software Development
Figure 1 - Input worksheet for Scrum simulation
These inputs allow a simulation model to be would allow specification of an exact
built. The calculations required at each step probability curve for random numbers, and
are pretty simple, except for the random this is exactly what commercial products
number generation (and this is also pretty offer. For this example, we stay within the
simple in Excel). A thorough explanation of simple but often indicative standard bell
strategies for building random numbers curve (Normal Distribution) with the user
that follow certain patterns evident in real- being able to specify the bounds and the
life for a given input is covered later; For mean adjustment for each estimate size (the
now it is just important to understand that rationale being that the bigger the estimate
each random number provided by the size, the more variability, but it is wise to
random number generator will after many look at historical data and make this model
generations follow a bell-curve pattern in conform to a team’s estimation ability).
(more occurrences happen around the
Once a random number is generated for
chosen mean value, falling off either side),
each estimate size bin (the number of stories
with 45% below that mean, and 45% above
with that estimate size), this number is
that mean. Less than 5% occur above and
multiplied by the total story size of that
below the specified lower and upper
estimate, and these are summed. For
bounds. More advanced tuning of a model
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Introduction to Monte-carlo Analysis for Software Development
example, Equation 1 shows the function to =((Total_Points/IntroducedScale)*In
troducedPerScale)*PointsPerIntroduc
equate the random number within the
ed
bounds chosen, and multiplying the total
story size for that estimate bin in order to Equation 3 – Calculating the adjustment to add for
introduced scope
get a total story points scenario for a single
simulation event. The project story point total, the defect
point total and the introduced scope total
=NORMINV(RAND(),Mean1,(UpperBoun
are summed and this value represents the
d1-LowBound1)/3.29)*Bin1Total
total story points to burn down over the
Equation 1 – Calculating the adjusted story points course of a project to achieve 100%
for an estimate size using a random number within
the boundaries chosen
complete. To determine the number of
iterations it would take to complete these
The equation shown in Equation 1 is stories, it is a matter of simply dividing by
replicated for each estimate size bin, and one of the three target points per iteration
these are summed to obtain a final “total” inputs as shown in Equation 4.
number of story points in an entire backlog.
=(Total_Points + Defect_Total +
Additional story points to account for Increased_Scope_Total) /
defects are calculated and added to the (PointsPerIteration*Vacation_Adj
total. The total project points is divided by ustment)
the number of defects per point input, and Equation 4 - Calculating the number of iterations
multiplied by the number of points per required to complete ALL points. In this model, we
calculate this for three different target point per
defect input, as shown in Equation 2 iteration inputs
=((Total_Project_Points/DefectSc Some adjustment is done at this point to
ale)*DefectsPerScale)*PointsPerD
account for developer vacation time, a
efect
particular problem for long running projects
Equation 2 – Calculating the adjustment to add for with large number of developers. An
defects
adjustment is calculated that reduces the
Scope is often added after the project number of point per iteration to account for
begins, whether in the form of new features, this. In this model, it is simply the formula
or work relating to fixing production shown in Equation 5.
defects, or non-development specific tasks.
=1-(AvgDevsOnVacation/TotalDevs)
This model simply applies a single
introduced stories value following the rate Equation 5 - Simple Vacation Adjustment Equation,
normally in the range of 0.9 to 0.95
specified as an input and adds this to the
total story points so far totaled. The above equations are run many
thousands of times in different rows
(simply the first row in copied down in
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Introduction to Monte-carlo Analysis for Software Development
Excel to get a set of simulation results). The of workdays, and optionally excluded
more times they are calculated, the firmer public holidays as shown in Equation 6
probability patterns will emerge. Figure 2 demonstrates.
shows the first five rows of many
=WORKDAY(StartingDate,
thousands. Each row will determine how DaysPerIteration *
many iterations it will take to complete a Iteration_Target,
backlog for the three target velocities, in this Public_Holiday_Dates)
example 190, 200 and 220 points per
Equation 6 - Finding the date given number of
iteration. workdays. Iteration_Target will be 7 to 12 for our
example. StartingDate and DaysPerIteration are user
The only remaining steps to determine inputs as shown in Figure 1
completion date and probability results like
To calculate the percentage probability of
those shown in Figure 3 is to calculate the
achieving a result at a target velocity (one of
calendar date, and how many simulation
three), the equation shown in Equation 7 is
rows fall within a given number of
used. This equation counts the number of
iterations. The Figure 3 results ask the user
simulation rows less than the target, divides
to give a range of iterations, seven to twelve
that by the total number of simulations to
in this example. The completion date is
find the percentage likelihood. This is done
calculated using a convenient Excel function
for each of the target velocities.
that determines a date from a given number
Figure 2 - The results for the calculations showing the first 5 simulations of many thousands
Figure 3 - The results showing probabilities of hitting certain dates
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Introduction to Monte-carlo Analysis for Software Development
=(COUNTIF(Velocity1Range,"< " & Kanban Model
Iteration_Target) Modeling a Kanban project using Excel is
/COUNT(Velocity1Range))
difficult, not because the calculations are
Equation 7 - Count the number of simulations that complex, but because the interaction
complete within a target, and convert to a between stories would require at least one
percentage
column per story, per simulation row, and
The results shown in Figure 3 indicate that this just gets un-maintainable. A custom
as long as the team can maintain 200 story application makes more sense, and this
points per iteration, they have an 87% article covers one such application.
chance of finishing by 22 October 2009
Kanban divides the steps of delivering a
(when this simulation was done). As a
single story into columns (called Status’
project progresses, the model can be tuned
throughout this article). For example, a
to improve confidence and accuracy. Defect
story might pass from Design, to
counts can be determined from the bug
Development, to Testing, to Release. The
tracking database (how many point for x
time taken for each story in each Status is
number of defects raised), the random
recorded. Work is limited in each Status,
number boundaries for each estimate size
and a new story can only be pulled from left
match actual prior data. By maintaining this
to right when a vacant position is available
model, the probability of hitting a given
(total cards within a Status are below the
date is always available, and some rigor
limit). A card system on a wall using post-it
was used in the calculation.
notes (or electronic version) is used to
Figure 4 - Example digital Kanban Board visualizing work flowing from left to right through a process.
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Introduction to Monte-carlo Analysis for Software Development
represent stories flowing from left to right below the WIP limit for that status. This
as shown in Figure 4. process continues until all cards have
traversed from the imaginary backlog to the
To simulate, the application takes the inputs
completed stories pile, and the time take to
of the number of Status columns, and a
do this is recorded.
lower bound and upper bound for time
taken to complete stories in each status, and This type of simulation avoids having to
the limit of stories allows in each status at have accurate estimates for each story by
one time (called the WIP Limit or work in looking at the previous lower and upper
progress limit). In place of an actual bounds for completing stories and using
backlog, a number of initial story cards are random numbers between these
specified by the user as shown in Figure 5. boundaries. It would be a small
These inputs are enough to do a simple enhancement to add the ability to have size
simulation, where the application loops for each story, but this would complicate
simulating a given time interval, for the model and may not increase accuracy in
example 1 day. The simulator grabs the first a significant way; The actual times
few stories and populates the first status measured on previous work is likely more
column. For each story a random time indicative of future patterns. These actual
within that status’ boundaries is calculated ranges can be mined from any work
and stories are only move to the right when tracking tool, and are often easy to read
a) that time has elapsed, b) there is an open from a Cumulative Flow Diagram which is
position that keeps the number of stories a graphical representation of how many
Figure 5 - Kanban Simulation Setup Screen for the basic inputs.
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Introduction to Monte-carlo Analysis for Software Development
cards are in each status at any given system where defects can be raised in
moment. different status’ and those defects will cause
a story to start back in another status for a
Defects, added scope and the time stories
random time between the specified
spend in a “Blocked” state (no test
boundaries.
environment, questions to a stakeholder,
un-available experts) are represented by
adding more stories to the backlog
according to rates specified by the user, and
extending story times by given user rates.
Each of these real-world values can be
obtained from tracking systems, the defect
database for example, or the spreadsheet
holding the story data, or initially guessed
from prior experience. Tuning these values
Figure 6 - Blocking rate, Defects, and Added scope
over time and demonstrating to the entire all materially impact time and need to be simulated
team the impact of these occurrences is a
Kanban simulation is carried out with the
great way to manage scope creep and
specified setup either visually for a single
quality issues in a team. Figure 6 shows
pass, or many hundreds or thousands of
how defects are specified in our example
times for Monte-carlo results. Once the
Histogram - Completion Date Probability
(Project start: 5/24/2011)
250
200
Frequency
150
100
Frequency
50
0
Completion Date
Figure 7 - Sample histogram of Monte-carlo simulated completion dates from a Kanban model
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Introduction to Monte-carlo Analysis for Software Development
simulation has run, this application writes as the results show in Figure 8.
the results to Excel for further analysis as
Monte-carlo simulation offers advantages to
shown in the histogram chart in Figure 7.
teams expected to give completion dates for
There is no absolute result, just a pattern of
projects, and to model the uncertainties in a
the most commonly simulated completion
productive way. Whilst Monte-carlo
dates, in this case early December to mid-
simulation doesn’t give an exact date, it
December is the likely range.
shows the likely pattern and ranges that can
Kanban simulation can answer another key be expected, and the factors that influence
question – If you had to add staff, how that date most, a process called Sensitivity
many and what skills do you need? If we Analysis.
assume that each Kanban status has a
specific skillset, for example, graphic Sensitivity Analysis – What input
designers in the Design status (Status 2 in has the most impact on date
this example), Developers in the Dev status Sensitivity analysis answers the question of
(Status 3), QA in the Testing status (Status what input factor has the greatest impact on
4), and release management represented in the final result. In essence, if all the inputs
the DevOps status (Status 5) – then by were increased and decreased by 10% (a
systematically increasing and decreasing consistent amount, 10% makes the math
the WIP limits for each status and executing easy), one at a time and a simulation run
a Monte-carlo simulation run, the status each time – how much each change
that has the most impact can be determined. impacted the final result.
The example simulator supports this feature
Figure 8 - Kanban simulation finds what Status column increase gives the best improvement.
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Introduction to Monte-carlo Analysis for Software Development
Figure 9 - Manual sensitivity analysis. Defects have more impact on average iterations than increased scope in
our example Scrum model. Motivate the team to reduce defect rate anyway they can.
Commercial Monte-carlo simulation tools Relevant Random Number
make this functionality easy to visualize in Generation
graphs, but for our Excel model, it is easy to Random number generation is a complex
do by hand. To determine if defects or field of mathematics. For truly random
introduced scope is having a bigger impact numbers, a computer is the last thing you
on outcome, temporarily increase the defect want. Random numbers generated by
rater and then the introduced scope rate by computer are never truly random, they rely
a percentage and take the average number on algorithms that attempt to be random,
of iterations before and after the change. but the algorithm used is repeatable given
Figure 9 shows such a result for the Scrum the same starting value, therefore – not
model used earlier. Although close, random! For most purposes this won’t
increasing the defect rate by a percentage cause an issue for the modeling we
has more impact on average number of undertake, but it is important to realize that
iterations to complete than the same random number generators have flaws, and
percentage rate change for additional scope. to avoid them if they will impact the results.
From observing many models, this is a
common case, and the model has helped To simulate effectively, we need sets of
teams understand the impact of quality random numbers that fall within the likely
earlier when developing code. After bounds of the real world problem. Excel
reducing defect rate, look for the next most helps with the function: Rand(). Rand()
important factor and improve that area. returns random numbers from 0 to 1 with
Sensitivity analysis and a Monte-carlo an equal chance of occurrence across those
simulation give teams the tool to bounds as shown in Figure 10, but
demonstrate how little improvements obtaining a random number within a bound
count. is just one part of the problem. In the real
world, the random numbers between a
range might occur more frequently around
one value, or end of the boundaries. If we
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Introduction to Monte-carlo Analysis for Software Development
ignored this, we compromise accuracy in Excel supports applying bias to the random
the final result. For example, when looking number generation, and custom
at the actual time taken for previous applications take this to a whole new level
estimates, a bias towards overrunning time offering features not only to produce sets of
could be the pattern. Even though the random numbers that fit a curve, but also to
boundaries might be from 80% (20% under look at existing data and match a random
the estimate) to 200% (double the estimate), number set to that data.
the majority of estimates are 175%. Left
In the Scrum model covered in the article,
alone, the random number generator in
we forced the random number generator to
Excel would evenly distribute random
follow the common Bell Curve, or Normal
values across the range.
Distribution as shown in Figure 11.
=RAND()
80
60
Frequency
40
20
Frequency
0
0.000301328
0.120246911
0.180219703
0.240192494
0.300165286
0.360138077
0.420110869
0.540056452
0.600029243
0.660002035
0.719974826
0.779947618
0.839920409
0.899893201
0.959865992
0.06027412
0.48008366
Bin
Figure 10 - Excel's RAND() function returns random numbers from the range 0 to 1 with equal probability
=NORMINV(RAND(),1,1)
200
Frequency
150
100
50
0 Frequency
0.447206008
0.947231395
1.447256781
1.947282167
2.447307553
3.447358326
3.947383712
4.447409099
4.947434485
-2.55294631
-3.052971696
-2.052920923
-1.552895537
-1.052870151
-0.552844764
-0.052819378
2.94733294
Bin
Figure 11 - To obtain random numbers that fit the Normal distribution (Bell curve), use the NormInv() function
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Introduction to Monte-carlo Analysis for Software Development
Figure 12 - EasyFit from Mathwave Technologies is a commercial curve fitting tool that can create random
numbers that fit real-world data
“Normal” is one distribution curve of without disrupting development staff for
many, and commercial Monte-carlo detailed analysis. Where possible, it is
packages support more than the basic recommended to analyze prior data, or to
curves. carefully consider the likely range of
possible values, and whether they are
EasyFit is a commercial curve-fitting
weighted more frequently towards one
package that will analyze existing data and
boundary than another when choosing a
determine what probability curve fits that
random number distribution fit. If it is
data. This application will also then create a
significant, then look for commercial tools.
set of random numbers that match this
curve, allowing you to simulate with a Conclusion
random input that is indicative of the real This article touches the surface of how to
world values. One use for this type of build and model software projects using
application is looking at the frequency of Monte-carlo techniques. The ability to
previous estimates and employing those quickly forecast a projects most likely
values in future Monte-carlo simulations. completion date, and the impact of adding
Figure 12 shows an actual set of estimates more staff, or reducing defect counts makes
from a previous project. Without any other this analysis an important tool for any
better information, random numbers development managers arsenal, and place
generated from this curve could be used to them in a position to answer with a level of
simulate future similar project estimates
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Introduction to Monte-carlo Analysis for Software Development
unprecedented confidence the three likely
ongoing upper-management questions -
1. How much will this product cost to
develop and deliver?
2. What is the likelihood of hitting date
x?
3. What resources do you need to hit
date x (money equals people, so the
question is often how much more
money do you need to hit date x)?
[END]
About the Author
Troy Magennis is founder of Focused
Objective, a consulting firm that aims to
improve software development practices
and management through better tools and
education. Troy has held positions at VP
level for many companies in diverse field
from Automotive, Financial, Image Rights
Management and Travel.
For feedback, Troy can be contacted at –
Troy.magennis@FocusedObjective.com
For more articles like this, visit use at –
http://www.FocusedObjective.com
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