The Stochastic Capacity Constraint

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The Stochastic Capacity Constraint
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MIDTERMNEW DATE AND TIME AND PLACE

• Tuesday, November 1
• 8pm to 9pm
• Woodsworth College
• WW111

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Estimates

• Estimates are never 100 certain
• E.g, if we estimate a feature at 20 ECDs
• Not saying will be done in 20 ECDs
• But then what are we saying?
• Are we confident in it?
• Is it optimistic?
• Is it pessimistic?
• A quantity whose value depends upon unknowns (or
  upon random chance) is called a stochastic
  variable
• Release planning contains many such stochastic
  variables.

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Confidence Intervals

• Say we toss a fair coin 5000 times
• We expect it to come up heads ½ the time 2500
  times or so
• Exactly 2500?
• Chance is only 1.1
• 2500?
• Chance is 50
• If we repeat this experiment over and over again
  (tossing a coin 5000 times), on average ½ the
  time it will be more, ½ the time less.
• 2530?
• Chance is 80
• 2550?
• Chance is 92
• These (50, 80, 92) are called confidence
  intervals
• With 80 confidence we can say that the number of
  heads will be less than 2530.

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Stochastic Variables

• Consider the work factor of a coder, w.
• When estimating in advance, w is a stochastic
  variable.
• Stochastic variables are described by statistical
  distributions
• A statistical distribution will tell you
• For any range of w
• The probability of w being within that range
• Can be described completely with a probability
  density function.
• X-axis all possible values of the stochastic
  variable
• Y-axis numbers gt 0
• The probability that the stochastic variables
  lies between two values a and b is given by the
  area under the p.d.f. between a and b.

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PDF for w

• Probability that 0.5 lt w lt 0.7 66
• Looks to be fairly accurate.
• Has a finite probability of being 0
• Has not much chance of being much greater than
  1.2 or so
• Drawing such a curve is the only real way of
  describing a stochastic variable mathematically.

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Parameterized Distributions

• So, Bill, heres a piece of paper, could you
  please draw me a p.d.f. for your work factor?
• Nobody knows the distribution to this level of
  accuracy
• Very hard to work with mathematically
• Usual method is to make an assumption about the
  overall shape of the curve, choosing from a few
  set shapes that are easy to work with
  mathematically.
• Then ask Bill for a few parameters that we can
  use to fit the curve.
• Because we are not so sure on our estimates
  anyways, the relative inaccuracy of choosing from
  one of a set of mathematically tractable p.d.f.s
  is small compared to the other estimation errors.

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e.g., a Normal for w

• Assume work factors are adequately described by a
  bell-shaped Normal distribution.
• 2 points are required to fit a Normal
• E.g., average case and some reasonable worst
  case.
• Average case half the time less, half the time
  more 0.6
• Worst case 95 of the time w wont be that bad
  (small) 0.4
• Normal curves that fits is N(0.6,0.12).

area 68
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Maybe not Normal

• Normals are easiest to work with mathematically.
• May not be the best thing to use for w
• Normal is symmetric about the mean
• E.g., N(0.6,0.12) predicts a 5 best case of
  0.8.
• What if Bill tells us the 5 best case is really
  1.0?
• Then cant use a Normal
• Would need a skewed (tilted) distribution with
  unsymmetrical 5 and 95 cases.
• Normal extends to infinity in both directions
• Finite probability of w lt 0 or w gt 10

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Estimates

• Most define our quantities very precisely
• E.g., for a feature estimate of 1 week
• Post-Facto
• What are the units?
• 40 hours? Longer? Shorter? Dedicated? Disrupted?
  One person or two? ...
• Dealt with this last lecture in great detail
• Stochastic
• 1 week best case?
• 1 week worst case?
• 1 week average case?
• Need a p.d.f
• Depending upon these concerns, my 1 week maybe
  somebody elses 4 weeks.
• Very significant issue in practice

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The Stochastic Capacity Constraint

• T is fixed
• F and N are both stochastic quantities.
• Can only speak about the chance of the goo
  fitting into the rectangle
• Say F400, N10, T40 are we good to go?
• Cannot say.
• Need precise distributions to F and N to answer,
  and then only at some confidence level.

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Summing Distributions

• F and N are sums and products over many
  contributing stochastic variables.
• E.g.
• F f1 f2
• If f1 and f2 have associated statistical
  distributions, what is the statistical
  distribution of F?
• In general, no answer.
• Special case f1 and f2 are both Normal
• Then F will be Normal as well.
• Mean of F will be the sums of the means of f1 and
  f2
• Standard deviation of F will be the square root
  of the sums of the squares of the standard
  deviations of f1 and f2.
• How about f1 f2?
• Figet about it! Huge formula, result is not a
  Normal distribution
• One needs statistical simulation software tools
  to do arithmetic on stochastic variables.

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Law of Large Numbers

• If we sum lots and lots of stochastic variables,
  the sum will approach a Normal distribution.
• Therefore something like F is going to be pretty
  close to Normal.
• E.g., 400 features summed
• N will also be, but a bit less so
• E.g., 10 ws summed

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Delta Statistic

• D(T) N ? T ? F
• If we have Normal approximations for N and F, can
  compute the Normal curve for D as a function of
  various Ts.
• We can then choose a T that leads to a D we can
  live with.
• Interested in
• Probability D(T) ? 0
• The probability that all features will be
  finished by dcut.
• In choosing T will want to choose a confidence
  interval the company can live with, e.g., 80.
• Then will pick a T such that D(T) ? 0 80 of the
  time.

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Example Picking T
confidence level confidence level confidence level confidence level confidence level confidence level confidence level
25 40 50 60 80 90 95
30 -39 -77 -100 -123 -177 -217 -250
35 14 -26 -50 -74 -130 -172 -207
40 67 25 0 -25 -84 -128 -164
T 45 121 77 50 23 -38 -85 -123
50 174 128 100 72 7 -41 -82
55 228 179 150 121 52 1 -41
60 282 231 200 169 97 44 0

• F is Normal with mean 400 and 90 worst case 500
• N is Normal with mean 10 and 90 worst case 8
• Cells are D(T) N ? T ? F at the indicated
  confidence level
• Note transitions through 0.

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Choices for T

• To be 95 certain of hitting the dates, choose T
  60 workdays
• Or... If we plan to take 40 workdays, only 5 of
  the time will be late by more than 20 workdays
• To be 80 sure, T 49
• To gamble, for a 25 fighting chance, make T 33.

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Shortcut

• Ask for 80 worst case estimates for everything.
• If F NxT using the 80 worst case values, then
  there is an 80 chance of making the release.
• The Deterministic Release Plan is based on this
  approach.
• If you also ask for mean cases for everything,
  can then fit a Normal distribution for D(T) and
  can predict the approximate probability of
  slipping.

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Initial Planning

• Start with a T
• Choose a feature set
• See if the plan works out
• If not, adjust T and/or the feature set an
  continue

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Adjusting the Release Plan

• Count on the w estimated to be too high and
  feature estimates to be too low.
• Re-adjust as new data comes in.
• Can pad the plan by choosing a 95 T.
• Will make it with a high degree of confidence
• May run out of work
• May gold plate features
• Better to have an A-list and a B-list
• Choose one T such that, e.g.,
• Have 95 confidence of making the A list
• Have 40 confidence of making the AB list.

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Appreciating Uncertainty

• Successful Gamblers and Traders
• Really understand probabilities
• Both will tell you the trick is to know when to
  take your losses
• In release planning, the equivalent is knowing
  when to go to the boss and say
• We need to move out the date
• Or we need to drop features from the plan

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Risk Tolerance

• Say a plan is at 60
• Developer may say
• Chances are poor 60 at best
• An entrepreneurial CEO will say
• Looking great! At least a 60 chance of making
  it.
• Should have an explicit discussion of risk
  tolerance

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Loading the Dice

• Can manage to affect the outcome.
• Like a football game
• Odds may be 3-to-1 against a team winning
• But by making a special effort, the team may
  still win
• In release planning
• Base the odds on history
• But as a manager, dont ever accept that history
  is as good as you can do!
• E.g., introduce a new practice that will boost
  productivity
• Estimate will increase productivity by 20
• Dont plan for that!
• Plan for what was achieved historically.
• Manage to get that 20 and change history for
  next time around.

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Example Stochastic Release Plan

• Sample Stochastic Release Plan