The Stochastic Capacity Constraint
CSC444F'05
Lecture 5
1
MIDTERM
NEW DATE AND TIME AND PLACE
Tuesday, November 1
8pm to 9pm
Woodsworth College
WW111
CSC444F'05
Lecture 5
2
Estimates
• Estimates are never 100% certain
• E.g, if we estimate a feature at 20 ECD’s
– 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.
CSC444F'05
Lecture 5
3
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.
CSC444F'05
Lecture 5
4
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 >= 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.
CSC444F'05
Lecture 5
5
PDF for w
• Probability that 0.5 < w < 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.
3
Probability density function for wi.
area = 0.66
2
1
0
1
0.6
0.5
CSC444F'05
2
3
0.7
Lecture 5
6
Parameterized Distributions
• “So, Bill, here’s 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.
CSC444F'05
Lecture 5
7
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 won’t be that bad (small) = 0.4
• Normal curves that fits is N(0.6,0.12).
 = 0.6
 = 0.12
N(0.6,0.12)
area = 68%
area = 0.95
0.4
CSC444F'05
0.6
Lecture 5
8
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 can’t 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 < 0 or w > 10
 = 0.6
 = 0.12
N(0.6,0.12)
area = 0.95
0.4
CSC444F'05
0.6
Lecture 5
9
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
else’s 4 weeks.
– Very significant issue in practice
CSC444F'05
Lecture 5
10
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 F=400, N=10, T=40: are we good to go?
– Cannot say.
– Need precise distributions to F and N to answer, and then only at some
confidence level.
CSC444F'05
Lecture 5
11
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.
CSC444F'05
Lecture 5
12
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 w’s summed
CSC444F'05
Lecture 5
13
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 T’s.
• 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.
CSC444F'05
Lecture 5
14
Example Picking T
confidence level
T
•
•
•
•
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
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.
CSC444F'05
Lecture 5
15
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.
CSC444F'05
Lecture 5
16
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.
CSC444F'05
Lecture 5
17
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
adjust T
no
adjust feature set
choose T
CSC444F'05
choose feature set
Lecture 5
yes
happy?
done
18
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 A+B list.
CSC444F'05
Lecture 5
19
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
CSC444F'05
Lecture 5
20
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
CSC444F'05
Lecture 5
21
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, don’t 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%
Don’t plan for that!
Plan for what was achieved historically.
Manage to get that 20% and change history for next time around.
CSC444F'05
Lecture 5
22
Example Stochastic Release Plan
• Sample Stochastic Release Plan
CSC444F'05
Lecture 5
23