Problem:
I run a pizza parlor which serves the local area. The approximate service area is a circle around the store. Because of this, and because housing density and ordering habits are pretty constant within this area, and because travel time is approximately linear with distance, I expect that on average, my journey time to deliver to a customer should be about $\frac{2}{3}$ of the time to travel from my shop to the edge of my service area. (As per Average distance from center of circle to evenly-distributed points within it). Furthermore, I expect the distribution of travel times to look approximately like this (scale irrelevant):
However, when I look at my actual times, they look more like this:
Theories on why such a discrepancy and arguments for the best distribution to describe the actual data? (Gamma/Lognormal perhaps).
There quite a few different possibilities that can explain the result. Is the distribution of customers really homogenous over the circular area or not? Most likely this is not the case. Is the traveling time really only linear dependent on the distance? This depends on the layout of the road network and is again usually not the case. In addition, the traveling time will depend on traffic, and hence will be different for different periods during the day due to traffic jams, rush hour, etc... What is the delivery scheme? You probably do not deliver one order at a time but combine a number of them before returning to the shop. How do you measure those delivery times an enter them in the charts. Having many deliveries at large distances, will take quite a bit of time and can cause all sorts of logistics problems.
From the given information, the real reason cannot be inferred. But in order to get some insight, you would have to make other plots. The delivery time is not convenient, because it already destroys some of the information. Much more useful is to make a distribution of the location customers on the city map in order to see whether this homogenous or that there are white spots. If so where are they? What are they correlated to? Other businesses or restaurants?
The next step would be to look at this info as function of the time over shop hours. Again, there might be various reasons for correlations. The good old pizza after couple of beers for instance or after a game.
You can also make a map of the delivery time attached to the location of customers with a colour-coded map. This can reveal the correctness of the assumption that delivery time is proportional to distance or you will find that certain areas are more difficult to reach than anticipated, and this might again also depend on the time of day.
These are just a few suggestions that might help you to understand better what goes on.