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Author Topic: Simple formula for rig lifetime total BTC output  (Read 2582 times)
marcus_of_augustus
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May 23, 2011, 10:30:44 PM
 #1

Just realised there is simple formula for estimating the total amount of BTC a given size rig will output.
We do need to make one assumption, estimate how large the average difficulty increase will be over the life of the rig.

So with this we can use the same formula that gives the total amount of BTC will be mined; (Bernoulli's summation of infinite geometric series formula)

http://forum.bitcoin.org/index.php?topic=7638.msg112479#msg112479

a + ar + ar2 + ar3 + ... + ark + ... = a/(1-r)

Eg; say your rig is currently producing 10 BTC per day, then that is around 140 BTC for the current difficulty period. This is the initial rig output, a,

a = 140 ; initial rig output of current difficulty period

Now a difficulty increase of say 50% is going to decrease your output in the next difficulty period by a third or

r = 1/1.5 = 2/3 ; 1/(difficulty multiplier)

So assuming an average difficult increase of 50% for life of rig, a machine earning 140 BTC in current period can be expected to output a total of

140/(1-0.6666) = 420 BTC

Let's try something a little more optimistic, say average difficulty increases of 30%; (r = 1/1.3)

140/(1-(1/1.3)) = 607 BTC

So;

a/(1-r) ; formula for total BTC over life of rig

where
a  ; initial rig output during current difficulty period
r  ; inverse of average difficulty multiplier (empirical r = 1/1.3)


Edit: chodpaba has done some pretty in-depth analysis of difficulty-price correlations so may have quite accurate number for average difficulty increase estimate.
Edit2: found chodpaba's data and calculated average difficulty increase over last 25 periods (50 weeks) to be 30.3%, see below.

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marcus_of_augustus
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May 23, 2011, 10:45:53 PM
 #2

As an aside let's do that same formula for the total installed capacity of today's network;

Current output, 7200 per day or approx. 100,800 BTC for current diff. period.

Assume average network increase of 30% for operating life of current installed capacity;

Total BTC mined by today's network over it's life = 436,800 BTC


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May 24, 2011, 11:37:07 AM
 #3


Difficulty increase multiplier (averaged over the last 25 difficulty periods) comes out to

1.3034498545   (i.e. 30.35%)

from chopaba's data set

https://spreadsheets.google.com/ccc?key=0AkzCs5F7cezpdHRCWjZIQ0haWmw0cHVKV0d6M2VpM0E&hl=en&authkey=CKGr0YsE#gid=0

(Not a bad guess.)



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