How average handling time differ for different number of simultaneous tasks Thelma, Louise and other people are handling tasks. One person can have many tasks at the same time.
I have a table of handled tasks with +1000 rows:
Id | Start               | End                 | Person
1  | 2015-06-05 12:25:16 | 2015-06-05 12:30:56 | Thelma
2  | 2015-06-05 12:27:00 | 2015-06-05 12:30:00 | Thelma
3  | 2015-06-05 12:30:00 | 2015-06-05 12:40:00 | Louise
I like to find out how the average handling time differ for different number of simultaneous tasks.
I.e.: When Thelma have 3 tasks, avg handling time is 5 min. But when Thelma have 4 tasks, avg handling time is 5:50 min... and so on.
I cannot in the world figure this out.
 A: Your question is quite open, so you will need to develop a model and apply it. In this case, you will also need to evaluate if your model is good.
To give you an idea, here is a simple one:

Each task has a required total effort $e$, which is just a positive real number of Thelma-minutes, randomly distributed according to a Gamma distribution. (A Thelma-minute is the effort of a task Thelma can theoretically finish in one minute if working exclusively on it.)
For each person $x$ and each number $n$ of simultaneous tasks, there is an associated rate $w_x(n)$ of applied work per task. By definition, $w_\text{Thelma}(1) = 1$ Thelma-minute/minute.
Given a task of total effort $e$ executed by a given person in the interval $[a;b]$, then
$$ e = \int_a^b w(n(t))\,\mathrm{d}t,$$
where $n(t)$ is the number of simultaneous tasks handled by that person at time $t$. This can also be written as
$$ e = \sum_{n=1}^\infty w(n) \Delta t(n),$$
where $\Delta t(n)$ is the total time spent in the task while simultaneously handling other $n-1$ tasks.
Given that framework, we can now derive a series of equations from the experimental data to fit the parameters. There will be one equation for each task, and the parameters to be estimated are $w_x(n)$ plus the parameters of the task effort distribution. The estimation can be done via maximum likelihood.

Then you need to evaluate the quality of the model. It is ultimately determined by the success of its application. For instance, if its goal is to increase team productivity, productivity increase will be the ultimate measure of its success. If its goal is to play with the data, we are interested in how fun it is.
Here are some example points for theoretical analysis:


*

*We assume that, given multiple tasks, a worker will evenly divide its effort among them. Is that reasonable for the application? Could we, for instance, add parameters to the model such that the work applied to a task is determined by a function $w_x(n, r)$ taking as inputs both the number of simultaneous tasks and the task's rank within that set?

*We do not assume any relation between $w_x(n)$ for different values of $n$. Would it be reasonable to reduce the number of parameters by assuming that it fits a certain family of functions, such as $w_x(n) = (c_x n + d_x)^{-1}$?

*We do not assume any relation between $w_x(n)$ for different values of $x$. Would it be reasonable to assume that it is of the form $w_x(n) = p_x w(n)$, that is, that all workers follow the same profile except for a productivity factor?

*We assume that the tasks' total efforts all come from a single Gamma distribution? Could there be different kinds of tasks such that each would have its own parameters?

*We assume that the tasks' total efforts are independent. Could there be some correlation between the efforts of tasks appearing around the same time? Could we add parameters to model that?


For an experimental assessment of the model, you might be interested in the following:


*

*Does the model fit the data? Do the model assumptions pass hypotheses tests?

*Are the measurements based on the model sufficiently precise?


Note that all these theoretical and experimental questions are permeated by the common fundamental theme of the choice between having more or less parameters. When we add parameters to the model, it is easier to fit the data, but the estimates we obtain will be less precise. On the other hand, when we remove parameters, it is harder to fit, but we can estimate with less data.
