Consider a project composed of activities X and Y having equal length and performed in that order

Question

Both activities can be laddered such that the first half of each activity is the predecessor for its own second half, so X1 precedes X2 and Y1 precedes Y2. The first half of X1 is also the predecessor for Y1. If the laddering procedure were performed again activities X and Y would be divided in four parts each, with precedence working as it did in the first split. Derive an expression for the minimum length of the project as laddering increases.

Answer 1

The original length of the project is X+Y. When laddering is performed the first time, the project length becomes X1+Y1+Y2; this is because X2 can be performed concurrently with Y1. If the activities are divided again, the project length will be X1+Y1+Y2+Y3+Y4; because X2 can be performed concurrently with Y1, X3 can be performed concurrently with Y2, and X4 can be performed concurrently with Y3.

In general, the project can be viewed as the length of Y plus the waiting time while completing the length of sub-activity X1. As the number of iterations of laddering these activities becomes large, the length of sub-activity X1 becomes small, reducing project length to the length of activity Y.
Expressed mathematically where P is the project length:

Without laddering P = x + y
1st ladder P = + y
2nd ladder P = + y
Subsequent P = = y