{"id":43,"date":"2014-12-15T17:12:00","date_gmt":"2014-12-15T16:12:00","guid":{"rendered":"https:\/\/caipirinha.spdns.org\/wp\/delay-propagation-in-sequential-task\/"},"modified":"2022-10-09T14:47:39","modified_gmt":"2022-10-09T12:47:39","slug":"delay-propagation-in-sequential-task","status":"publish","type":"post","link":"https:\/\/caipirinha.spdns.org\/wp\/?p=43","title":{"rendered":"Delay Propagation in Sequential Task Chains"},"content":{"rendered":"

<\/span>\u200bProblem Statement<\/span><\/h2>\n

Today, we want to examine how delays in sequential tasks propagate if we have a discrete and non-binary probabilistic pattern of delay for each involved task. That sounds quite complicated, but such problem statements do exist in real life. Let’s look into the source data…<\/span><\/p>\n

<\/a><\/div>\n

 <\/p>\n

Source Data<\/span><\/h2>\n

In our example, we have 3 tasks, named “A”, “B” and “C”, and their respective durations is distributed according to the table and the graph below:<\/span><\/p>\n\n\n\n\n\n\n
Duration<\/span><\/b><\/td>\n0 d<\/span><\/b><\/td>\n1 d<\/span><\/b><\/td>\n2 d<\/span><\/b><\/td>\n3 d<\/span><\/b><\/td>\n4 d<\/span><\/b><\/td>\n5 d<\/span><\/b><\/td>\n6 d<\/span><\/b><\/td>\n7 d<\/span><\/b><\/td>\n8 d<\/span><\/b><\/td>\n9 d<\/span><\/b><\/td>\n10 d<\/span><\/b><\/td>\n11 d<\/span><\/b><\/td>\n12 d<\/span><\/b><\/td>\n13 d<\/span><\/b><\/td>\n14 d<\/span><\/b><\/td>\n15 d<\/span><\/b><\/td>\n16 d<\/span><\/b><\/td>\n17 d<\/span><\/b><\/td>\n18 d<\/span><\/b><\/td>\n19 d<\/span><\/b><\/td>\n20 d<\/span><\/b><\/td>\n <\/td>\n<\/tr>\n
A<\/span><\/b><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n5%<\/span><\/td>\n10%<\/span><\/td>\n25%<\/span><\/td>\n20%<\/span><\/td>\n15%<\/span><\/td>\n10%<\/span><\/td>\n10%<\/span><\/td>\n5%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n100%<\/span><\/td>\n<\/tr>\n
B<\/span><\/b><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n40%<\/span><\/td>\n30%<\/span><\/td>\n20%<\/span><\/td>\n10%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n100%<\/span><\/td>\n<\/tr>\n
C<\/span><\/b><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n50%<\/span><\/td>\n50%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n0%<\/span><\/td>\n100%<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/span><\/span> \"Probability<\/span><\/h1>\n

As you can see, there is not a fixed duration for each task as we have that in traditional project management. Rather than that, there are several possible outcomes for each task with respect to the real duration, and the probability of the outcome is distributed as in the graph above. The area below each graph will sum up to 1 (100%) for each task.<\/span>
\nIf we assume these distributions, how does the probability distribution of sequence of tasks A, B, C look like?<\/span><\/p>\n

Solution<\/span><\/h2>\n

Level the durations<\/span><\/h3>\n

In a first approach, we define a minimum duration for each tasks. This is not mandatory, but it helps us to reduce the memory demand in our algorithm that we will use in a later stage. While we can define any minimum duration from zero to the first non-zero value of the probability distribution, we set them in our example to:<\/span><\/p>\n