Sampling a Population of Unknown Size

Question

How do you sample a population whose size is not known?

For example, I have a day's population from whom I need to select 1000 items from.

Each hour (for one day) some number of new items (can be any number, even 0) enter a box (at any time, not necessarily the start of the hour) from which I can choose any n items for my sample. However, at the start of each new hour, the box is emptied.

The items I choose go into my sample box, which cannot hold more than 1000 items and which I cannot change once I commit an item to it. So, I have to make my decision of how many items to choose during that hour time period.

I want my final sample to be as representative of the day's total population (all items that have been in that box at some point during that day) as possible.

Thank you. Are there any formulas or algorithms out there for this sort of idea?

How would my answer change if I could put items back that I originally selected to be in my sample?

EDIT: I can access my samples from the previous days if necessary, but only to look at - not change.

Each item comes with a timestamp for when they entered the box.

Answer 1

Not a complete solution but a starting point that you can tweak.

Use past data to compute the average number of items that arrive per hour during each one of the 24 hours. Suppose, the average number of items are: $(n_1, n_2, \ldots, n_{24})$. Then in hour $i$ you should expect that about $\frac{n_i}{n}$ items arrive where $n=\sum_in_i$.

Therefore, if $t_i$ items arrive in hour $i$ you should choose about $t_i \frac{n_i}{n}$ items at random.