## Task backlog waiting times are power laws

Once it has been agreed to implement new functionality, how long do the associated tasks have to wait in the to-do queue?

An analysis of the SiP task data finds that waiting time has a power law distribution, i.e., , where is the number of tasks waiting a given amount of time; the LSST:DM Sprint/Story-point/Story has the same distribution. Is this a coincidence, or does task waiting time always have this form?

Queueing theory analyses the properties of systems involving the arrival of tasks, one or more queues, and limited implementation resources.

A basic result of queueing theory is that task waiting time has an exponential distribution, i.e., not a power law. What software task implementation behavior is sufficiently different from basic queueing theory to cause its waiting time to have a power law?

As always, my first line of attack was to find data from other domains, hopefully with an accompanying analysis modelling the behavior. It’s possible that my two samples are just way outside the norm.

Eventually I found an analysis of the letter writing response time of Darwin, Einstein and Freud (my email asking for the data has not yet received a reply). Somebody writes to a famous scientist (the scientist has to be famous enough for people to want to create a collection of their papers and letters), the scientist decides to add this letter to the pile (i.e., queue) of letters to reply to, eventually a reply is written. What is the distribution of waiting times for replies? Yes, it’s a power law, but with an exponent of -1.5, rather than -1.

The change made to the basic queueing model is to assign priorities to tasks, and then choose the task with the highest priority (rather than a random task, or the one that has been waiting the longest). Provided the queue never becomes empty (i.e., there are always waiting tasks), the waiting time is a power law with exponent -1.5; this behavior is independent of queue length and distribution of priorities (simulations confirm this behavior).

However, the exponent for my software data, and other data, is not -1.5, it is -1. A 2008 paper by Albert-László Barabási ( detailed analysis)showed how a modification to the task selection process produces the desired exponent of -1. Each of the tasks currently in the queue is assigned a probability of selection, this probability is proportional to the priority of the corresponding task (i.e., the sum of the priorities/probabilities of all the tasks in the queue is assumed to be constant); task selection is weighted by this probability.

So we have a queueing model whose task waiting time is a power law with an exponent of -1. How well does this model map to software task selection behavior?

One apparent difference between the queueing model and waiting software tasks is that software tasks are assigned to a small number of priorities (e.g., Critical, Major, Minor), while each task in the model queue has a unique priority (otherwise a tie-break rule would have to be specified). In practice, I think that the developers involved do assign unique priorities to tasks.

Why wouldn’t a developer simply select what they consider to be the highest priority task to work on next?

Perhaps each developer does select what they consider to be the highest priority task, but different developers have different opinions about which task has the highest priority. The priority assigned to a task by different developers will have some probability distribution. If task priority assignment by developers is correlated, then the behavior is effectively the same as the queueing model, i.e., the probability component is supplied by different developers having different opinions and the correlation provides a clustering of priorities assigned to each task (i.e., not a uniform distribution).

If this mapping is correct, the task waiting time for a system implemented by one developer should have a power law exponent of -1.5, just like letter writing data.

The number of sprints that a story is assigned to, before being completely implemented, is a power law whose exponent varies around -3. An explanation of this behavior based on priority queues looks possible; we shall see…

The queueing models discussed above are a subset of the field known as bursty dynamics; see the review paper Bursty Human Dynamics for human behavior related aspects.