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November 20th, 2009, 02:34 PM   #1
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Math Focus: primes of course
Numeric ID systems

OK, this is a question relevant to an area of my interest. I will try to adequately couch it, my questions are at the end.

For illustration purposes, I can use the US social security system (though my particulars are something completely different).
The gist of the question is to know if a chosen numbering scheme set to a certain sizing is adequate for the system mechanics.

Note that as described here my system does differ from the actual SS one in several areas (re: reuse of numbers, no reserved/off limit numbers, no xtraneous content via any digit blocks)

Let's just specify some conditions:

initial population estim 350 million (now)
numbering sys 0-999999999, ie max is an unsigned 9-digit integer, approx 1 billion (US) useable numbers
numbers are assigned in sequence to live births, except to fill in gaps, as upon death, numbers can be reassigned.
Since assignment is strictly enforced, "collisions" are not possible (deaths are always processed before births)
We can presume number assignments are made once a day
I think we need a maximum birth/mortality ratio, say it is always <1 above a limit, say 500 million people
Overall, the maximum birth/mortality ratio ever seen is 1.05 ( I have no idea of actual numbers, or what pdf this ratio sees)
SO, if we have an adequate number pool, the total # of active numbers is fluctuating, and the highest in use is creeping up to some bound for a while (helped in part by the highest numbers being the youngest, longest-lived people generally).

Simple illustration
1,2,3,...3million (initial start)
say ID #20000 is the next change (death). We reuse the ID and still have 1,2,3,...3million
IF births xceed deaths, we might see a time progression like
1,2,3,...3000001 (0 deaths, 1 birth)
1,2,3,...3000002 (0 deaths, 1 birth)
1,2,3,...3000002 (1 death, 0 births, 1 intermed # unused/available)
1,2,3,...3000002 (2 deaths, 2 births, 1 intermed # unused/available still)
1,2,3,...3000004 (0 deaths, 3 births, dense-packed once again)
...etc

Q1) have I adequately asked/bounded the question. If not, please assist with what further info would be needed

Q2) if properly framed, how does one begin to setup for/prove the solution to "Is the numbering framework adequate without having to be xpanded?" (or if not, when is it xpected to max out?)

[having a no re-use rule reduces the problem complexity quite a bit, as for as estimating the "max out" date ]
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November 20th, 2009, 03:02 PM   #2
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Re: Numeric ID systems

The maximum number in use can be no greater than the greatest population in historical record. So as long as population can't grow beyond 1 billion, the system is adequate, and vice versa.
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November 20th, 2009, 04:50 PM   #3
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Re: Numeric ID systems

yes, that's the simple answer, which of course made me realize that the constraint I gave at 500000 would guarantee a bounded set ...
Besides the gross rate bounds, then it is only complicated when such simple bounds are not known, or can only be estimated.
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