The Erlang distribution has a straightforward interpretation in terms of waiting time for the occurrence of a predefined number of events in a Poisson process or a sum of a predefined number of exponential random variables. The gamma distribution is more general since it allows for a non-integer parameter, but it is usually given the same motivation. I know that this question was raised several times but I have not seen a satisfying answer so I will venture to pose it again: what is the canonical or at least a prototypical example of a random process that gives rise to some Gamma distributed random variable, which is not at the same time an Erlang random variable?
It seems you are asking for "real-life" examples where gamma distributions are used to model some real-world observables represented by random variables. There are many such examples. Take the Erlang distribution you mention first: The integer parameter case follows from some theoretical probability model for waiting times, but for modelling directly real-world waiting times, the gamma family with non-integer parameter will give better flexibility. Some other examples can be found here: Real-life examples of common distributions
Gamma distributions can model positive random variables, precipitation insurance (A quote from that paper Climatologists prefer the gamma distribution because it is sufficiently flexible to adequately characterize cumulative precipitation over time periods of varying length and link to a freely accessible version).
Other insurance use of gamma regression, in hydrology for modeling rainfall or floods, ... Inventory control use of the Gamma distribution, quote from that paper: In the field of inventory control of finished goods we find that the observed frequency distributions of demand have the following general characteristics:
- they exist only for non-negative values of demand
as the mean demand of items increases the observed distributions change from:
(a) monotonic decreasing to
(b) unimodal distributions heavily skewed to the right, and finally to
(c) normal type distributions (truncated at zero)
... and they then observe that the Gamma family of distributions matches nicely this qualitative behavior. This is an important point in modeling, we are not only interested in how a certain individual distribution matches a particular dataset, we are interested in the general behavior of a family of distributions.
The classic McCullagh/Nelder chapter 8 "Models for data with constant coefficient of variation" mostly uses the gamma distribution, gamma regression.
Given a radioactive sample with unknown emission rate $\lambda$, the likelihood on $\lambda$ induced by observations of emissions is gamma distributed.
Given a normal process with known mean, but unknown precision, the induced likelihood on the precision is gamma distributed.
And similarly for Pareto and gamma models.