Task Performance Data

about task performance data for XDCPM

Task performance data is a critical part of Earned Value Management and cost engineering.

Task performance is a measure of most any task-level system input or output; Sometimes called a dimension of performance, usually focus is on time (duration) and cost (monetary). Sustainable initiatives may include other inputs and outputs to estimate environmental impacts, such as by tracking greenhouse gas emissions or amount of fuel consumed.

XDCPM represents performance curves using discrete probability distributions somewhat akin to Inverse transform sampling and discrete uniform distributions and augmented by allowing for nonuniform distributions via digital sampling techniques.

1. Estimating using minimum, median, and maximum values.

Earned Value estimating sometimes uses other labels to indicate range of performance.

For duration, "optimistic", "most likely" and "pessimistic" (or "short", "median", "long") are labels commonly used by EV in place of minimum, median, and maximum.

For cost, "low", "most likely", and "high" are commonly used by EV in place of minimum, median, and maximum.

XDCPM converts minimum/median/maximum and other limited representations to discrete probability distributions so that network calculations are consistent. Min/Med/Max converstions transform a non-uniform signal discrete sampling representation of the normalized probability density function. The code includes comments on the process.

2. Estimating using discrete probability distributions

XDCPM uses ordered discrete probability distributions to represent probability curves for all cases. Each performance dimension (time, cost etc) has its own data that approximates a curve. A table of data consists of at least two columns: x and y; where x represents a probability of the measure of y; where y is the measured dimension (duration, cost etc.).

In probability calculations, each x is normalized by dividing the sum of all x in a distribution to arrive at a probability of x. This works whether x in a distribution represents probability or count or 1 for each individual task.

Data points are expected to be in strict order as if representing a continuous probability distribution.

Part of the magic of XDCPM is how it represents distribution curves as nonuniform discrete distributions with these definitions:

  • xi = probability = Δ percentage-point = a probability range in the distribution.
  • yi is a performance value in the distribution.
  • pp is a percentage-point in the distribution (0 to 1), where pp = 0 is the start of the distribution ie. end of left tail (standard minimum), 0.5 is the median, and 1 is the end of the distribution ie end of right tail (standard maximum).
  • One-to-one correspondence between x and y. For each xi there is a yi and visa versa.
  • The values of x and y may not be unique.
  • yx = F1( xy ) ; where yx is value of y for range of probability x, and xy is value of probability range x at performance value of y. In discussion, x and y subscripts are assumed for most cases.
  • yx = F2( pp ) where pp is a unique percentage-point when a discrete distribution represents a curve.
  • If n represents a range of percentage-points over one value of x, Δpp = xi = xn where i is a position of x in a distribution.
  • yn = yx for pp over a range of x ie n.
  • y = yn where set of all x is an ordered list from i=1 to m, and n is a minimum index of x satisfying the expression:
    the sum of x from 1 to n divided by sum of all x is greater than or equal to pp

The F2 is a function with a probability point argument from 0 to 1 --much like the percentage of circumference from a reference point of a spinner's circle. Percentage point range of 0 to 1 represents two standard deviations to the left and right of the median --four standard deviations altogether-- to approximate integration with existing uses in EV estimating and time expected calculations.

By keeping data in a sequential order, variations between dimensions keep their corresponding values. Although duration may consist of increasing y values as pp goes from 0 to 1; a corresponding fuel consumption may consist of decreasing y values for the same pp range indicating a common observation that fuel consumption efficiency is inversely proportional to duration to complete a task.

By traversing F2 at intervals of probability, a new probability distribution can be approximated from a sequence of other probability distributions.

XDCPM converts min/med/max values to a probability distribution by calculating the minimum to median and median to maximum probability tails separately. See the procedure acc_fin::pert_omp_to_normal_dc in the package documentation for code details. Regression tests are installed with each XDCPM via acs-automated-testing for calculations related to the XDCPM process and tests.