In 1960’s study made by Peter V Norden of IBM concluded that projects show a well defined pattern. Lter on Putenam verified Norden’s finding in software projects also and said that software development process follows Rayleigh shaped rate curve as and hence fits the mathematical formula for Rayleigh distribution.

Putnam father stated that the curve can be applied not only to the complete life cycle but also to the individual phases i.e., coding number of valid lines of code, errors found and corrected etc. According to Putnam relationship between size, schedule and effort matched the Norden/Rayleigh function which is given by following mathematical function.

         m(t) = 2 Kat (exp-at2)
where   (t) = number of persons at any time t (in years)
            K = Total project effort in staff years
            a = Acceleration factor (which determine sharpness of curve)

Acceleration factor is given by:

           a = 1/2 td²
Here   td = development time.

Rayleigh curve

Acceleration factor is given by:

Norden/Rayleigh function helped in understanding several other important characteristics/matrices of the projects. One such metric is Difficulty metric D which shows that when either manpower demand is very high or development time is short, project is difficult to develop. This value is computed by differentiating equation (1) and calculating value at t = 0. Difficulty metric is expressed as:

D = K/td²

Similarly peak manpower is given by:

m₀ = K/td^e1/2

From equation (2) and (3), Difficulty metric can be related to peak manpower as:

D = K/t2 = m₀^e1/2 / t_d

Another metric of interest is manpower buildup which can be found from equation (4) by taking the derivative w.r.t td and K respectively.

Manpower buildup D₀ can be expressed as:
D₀ = K/t_d³ person/year²

It was found that value of D0 is almost fixed for different type of projects and can have values equal to 8, 15 or 27 depending upon whether the project is a new software with many interfaces, a new stand along software or a software rebuilt from existing software. Depending upon the organization’s expertise the value of D0 can vary slightly from one company to another. The larger the value of D0, steeper will be the manpower distribution or in other words, manpower buildup wili be faster.

Putnam used data of various projects and results of Ryleigh shaped profile of software development, also derived a mathematical non-linear relationship between size of project, effort and time needed to complete it. It was called software equation. This equation makes use of a parameter which is a measure of productivity and is expressed as:

Productivity measure = Size / (Effort/B)^1/3 *(Time)^4/3

where effort is expressed in man years, time is years, size is effective source lines of code and B is special skills factor related to size. The values of B vary with size and its values are tabulated for different system sizes. Equation (6) can be rearranged to find effort as follows:

B * (Size)³ / (Time)⁴ * (Productivity measure)³

It is clear that effort required depends on size, time and productivity measures. The values of productivity measure are represented as Productivity Index (PI). Productivity Index can take values between 1 to 40 and corresponding productivity measure values are also tabulated. Low values of PI are used for highly complex projects or low productivity environments. Similarly hgh values of Pi indicate high productivity and well understood projects. By taking necessary measures, organizations can improve their PI as if makes lots of different in effort, cost and schedule of project.

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