The Shape of Code

When working on a project, what process do developers use to select the next task to implement?

One way to answer this question is to ask the developers/managers working on the project. However, these people are not always available, and sometimes the actual process used is not what management say it is.

Analysis of the amount of time a task spends in the queue finds various patterns. A previous post discussed some of the theory and data on the distribution of queue task waiting times.

What can be deduced about the task selection process from data on the time they spend in the queue?

From the theoretical perspective there are three task selection processes:

• select the task with the highest priority. The priority of a task might be set by its age (e.g., FIFO, a first, in first out queue), its value to the business, dependency of other tasks, etc.

  The distribution of the number of tasks having a given waiting time on some priority queues is a power law (in a FIFO queue the waiting time distribution depends on the distribution of arrival times of new tasks), i.e., , where is some constant

• select a task at random. Random selection comes in various guises including: developers picking what looks to be the most interesting task on the day, or managers deciding that particular functionality would look good during a marketing/sales pitch in a few days.

  The distribution of the number of tasks having a given waiting when tasks are selected at random from the queue is an exponential, i.e., , where is some constant,

• select a fraction, , of tasks by highest priority, and select the other fraction, , of tasks at random.

  The distribution of the number of tasks having a given waiting when tasks in this process are a combination of power law and exponential, i.e., , where and are constants.

The average amount of time a task spends in a queue is given by Little’s law, and is independent of the selection process (high priority tasks have a shorter waiting time than low priority tasks, but the overall average is unchanged), i.e., assuming that the averages, and variance, do not change over time, then: (waiting time) equals (average number of tasks in the queue) divided by (average number of tasks implemented per unit of time).

If analysis of the data finds just a power law, then the selection process involves a power law, just an exponential, then some random process(es), and if a combination distributions then a combination of selection processes.

Can anything be learned from knowing the value of or ?

The analysis of various priority+random models finds that, given enough running time, converges to specific values between 1 and 2. Is waiting time data on, say, 1,000 tasks (with an average of 4-hours per task, this is 2-person years) sufficient running time to converge to a good enough approximation of ?

This question can be answered by simulating task queue waiting times. The plot below shows, for 100 projects, the values of fitted to the power law component of the waiting time distribution of projects each having 1,000 tasks (the queue length was fixed at 10 tasks, five task priorities, with fraction of tasks selected by priority, and the rest randomly; code):

The wide range of fitted exponent values for clearly shows that 1,000 tasks is not sufficient for convergence to a good enough approximation. In fact over 100,000 tasks are needed before exponent values converge within one decimal place.

Similar results are obtained for queue lengths between 5 and 20 tasks, and priority ranges between 3 and 10. Reducing the value of down to 0.7 had a large impact on the tail of the distribution. I have not tried variable length queues.

Tasks that have spent a long time in the queue are likely to have their priority increased, or be removed from the queue.

If the amount of time spent in the queue contributes some amount of priority to the original task, then the waiting time distribution is exponential (technically it is geometric, the discrete form of exponential; code). The LLM maths assistants did not find any viable equations.

To summarise: The distribution of tasks having a given waiting time has a high variance, which significantly reduces its usefulness for deducing information about the task selection process.