Thank you very much for your help, everyone. 
To make it more clear what I am struggling with, I have written it down now as a library: SlidingWindow. (Still very work-in-progress; it does not have a lot of documentation yet as things might still change a lot.)
It does what I want to perform in a finite-state-automaton kind of way; that is, SlidingWindows does not care about GenServers or anything at all right now, but only about gathering and updating the information stored in the struct.
An example of a behaviour to calculate averages: (See also the file test_helper.exs)
defmodule TestSW do
@behaviour SlidingWindow.Behaviour
defmodule Transaction do
defstruct [:value, :created_at]
def new(value, created_at) do
%__MODULE__{value: value, created_at: created_at}
end
end
defstruct count: 0, sum: 0, product: 1
def empty_aggregate() do
%__MODULE__{}
end
def add_item(agg, %Transaction{value: int}) do
%__MODULE__{agg |
count: agg.count + 1,
sum: agg.sum + int,
product: agg.product * int
}
end
def remove_item(agg, %Transaction{value: int}) do
%__MODULE__{agg |
count: agg.count - 1,
sum: agg.sum - int,
product: div(agg.product, int)
}
end
def extract_timestamp(%Transaction{created_at: timestamp}) do
timestamp
end
end
And then you can create it using something like:
result =
SlidingWindow.init(TestSW, 10, Timex.Duration.from_seconds(2), initial_data())
|> SlidingWindow.shift_stale_items(ten_sec_future)
|> SlidingWindow.add_item(%Transaction.new(10, Timex.now())
|> SlidingWindow.add_item(%Transaction.new(15, Timex.now())
# And at some point:
|> SlidingWindow.get_aggregates()
# Result now contains something like:
result == %{0 => %TestSW{count: 0, product: 1, sum: 0},
1 => %TestSW{count: 0, product: 1, sum: 0},
2 => %TestSW{count: 0, product: 1, sum: 0},
3 => %TestSW{count: 1, product: 1, sum: 1},
4 => %TestSW{count: 2, product: 6, sum: 5},
5 => %TestSW{count: 2, product: 20, sum: 9},
6 => %TestSW{count: 2, product: 42, sum: 13},
7 => %TestSW{count: 2, product: 72, sum: 17},
8 => %TestSW{count: 2, product: 110, sum: 21},
9 => %TestSW{count: 2, product: 156, sum: 25}}
In the end, I think that the smartest decision I can make is to not mess with wrapping this inside of a GenServer-like layer myself, as there are an infinite amount of possible ways how someone might want to manipulate the FSA. Rather, I’ll write some example code on how it could be wrapped in your own GenServer including an send_after message to ensure that the data inside never becomes stale.
