Solving the fund manager dilema

Nassim Taleb pointed out a problem with assessing fund managers in The Black Swan: The Impact of the Highly Improbable. A certain number of fund manager's are going to do better than the market through dumb luck. Telling whether your fund manager is an exception or not is made more difficult by the fact that under performing managers are fired and new fund managers are added to the pool.

The math for static pool is straight forward: Start out with 1000 fund managers. Assess them at the end of the year. Fire the half that didn't beat the market. After the first year you have 500 survivors. After 2 years 250. After 3 years 125. And so on. If we let S be the number of survivors and P be the size of the initial pool of managers and t be the number of years, the equation would look something like this:

S = P(2)^-t

And of course you can easily generalize this to allow for different drop criteria by adding a factor ρ for the ratio being kept:

S = P(ρ)^-t

So what do you do when the size of the pool is changing? While I'm not directly solving the fund manager problem, I am solving an analogous one with books. Here is the scenario:

• A pool of electronic books is made available to library patrons
• When a library patron accesses a book the library pays a rental fee for a short term loan
• After a set number of loans the library purchases the book outright and the book remains as part of the library's collection
• Books are added to the pool as they become available
• Books are removed from the pool, for example a later edition is published and replaces the existing edition
• No purchased books are removed from the pool

How much money do you need to set aside to cover the costs of the rental fees and the purchase fees?
Which is really a bunch of related problems:

• How many books will I purchase? At what cost (% of list price)?
• How many books will I rent? How many times? At what cost?
• How many books will be removed from the pool before being purchased?

In each of these cases I can use the same equation except I need to measure the size of the initial pool, P, the number of survivors, S, at time, t, to calculate my rate, ρ. Measuring the number of survivors is easy - count the number of items that meet the selected criteria at time t. For example the number of books rented at least once after 100 days. I've been grappling with how to measure the size of the initial pool. What is the correct size for the initial pool? I think I have an answer:
count the number of items that have been in the pool at least that long and were not removed from the pool before reaching that age. So what do you think? Is that the right way to do this?