If you have two parallel activities preceding one, your project will be probably late. This is not rhetoric and certainly not ironic. It’s a mathematical fact.

Let’s take a plan of two activities ending at the same time. Let’s suppose the time estimation for the activities was made so that it’s equally probable to finish each activity before or after the estimated ending time t (the common way). The probability that both activities end in a moment less or equal to t, will be the product P( x <= t )P( y <= t), hence ½ * ½ = ¼. It means that it is improbable that activities end on time.

The most likely finish time can be calculated adding the mean duration to the start time. The duration is a non deterministic variable. It has a probability distribution that depends on the activity nature. Nevertheless, I’m going to show this simple fact doing mathematics with two activities with different exponential distributions starting at different times. The result can be seen in the following image. The two activities can be modeled as one. The most probable ending time of the composite activity can be obtained adding delta (δ) to the ending of the parallel activities.

What is the importance of this? This is a common problem doing plans. There are projects condemned to being late from the planning stage. Make the plan using feeding buffers as it is recommended by the people of “critical chain”. This is a good manner to avoid this problem. Of course you can ask consultants like me, to help you with these issues.

**Proof**
Let's calculate the most probable finish time of the composite activity.

The formula is:

The inferior limit d is because the function has zero value before the limit:

Replacing with the exponential probability functions:

Reordering:

Integrating:

The result is the finish of the activities plus delta (δ):