9.3 More on ASIC choices

0. Introduction
In the first ASIC choices post I derived some simple equations to allow you to determine which ASIC device would be a better choice depending on hashrate, difficulty, device price and local electricity costs. The downside of these equations is that while they can be used to determine the amount earned in a given difficulty period, to calculate the cumulative amount earned requires using a cumulative sum If we wish to rearrange the equation in terms of another variable - for example using initial difficulty or exchange rate as dependant variables, the cumulative sum causes difficulties.

With this in mind I derived an integral of the equation that is very close to the results of the step function D based cumulative sum, varying from it by only up to 0.01%. I will present the initial equations, and the integral here and leave the derivation as an exercise for the reader. I have renamed some variables for easier interpretation.

net profit  = cumulative sum (r*g*B/(D*p^(t-1))/p)*1e09/2^32*3600*24*14 - cumulative cost - device price

Where:

B = bitcoin mining block reward

g = device hashrate in Ghps

r = local exchange rate, fiat per btc

D = initial mining difficulty

p = assumed percentage increase in D per difficulty period

t = number of difficulty periods

cumulative sum (r*g*B/(D*p^(t-1))/p) ~ integral(r*g*B/(D*p^(t-1))/p)+constant

integral(r*g*B/(D*p^(t-1))/p) = -B*g*r/(D*log(p))*p^(-0.5)*(p^(-t)-1)*3600*24*14*1e09/2^32

So,  net profit is given by:

(-B*g*r/(D*log(p))*p^(-0.5))*(p^(-t)-1)*3600*24*14*1e09/2^32-ec/3600/1e03*dw*3600*24*14/p*t-dc

Where:

ec = electricity cost, local currency per kWh

dw = device wattage

dc = device cost, local currency

The advantage of equation is that we can now fairly simply rearrange it. After a discussion with bitcointalk.org forum member Jutarul and a comment by another member ldrgn, I decided to try some charts that a) showed a % ROI for a given device and b) indicated what the exchange rate would have to be in order to result in a particular ROI.

This addresses Jutarul's suggestion that an ROI type chart would be more intuitively understandable by most miners, and by making the exchange rate a dependant variable, ldrgn's belief that showing exchange rates much over US$40/btc is pointless. While on that subject, the reason for including such a wide range of US$/BTC prices was for comparative purposes only - in some localities and for some starting difficulties, the required exchange rate in order to break even at all is unlikely. In this case you'd know not to purchase that particular device.

In order to make the exchange rate a dependant variable, we simply rearrange the equation above :

r=-(ec/3600/1e03*dw*3600*24*14/p*t+dc+np)/(3600*24*14*1e09/2^32)/(p^(-t)-1)/p^(-0.5)*(D*log(p))/B/g

Where:

np = net profit

all other variables are as above

Then, varying only np and t we can calculate the required exchange rate and tile plot the results, as in chart 1 below.

Please remember that these charts are based on several assumptions; one or more of which are likely to be incorrect. The assumptions are:

1. Device specifications are correct

2. Addition of ASIC hashrate has reduced so that the average increase in D is 5% per difficulty period.

3. D at which you start mining with the new device is only assessed at 30, 60 and 90 million.

4. The local electricity cost is only assessed at US$0.1/kWh, US$0.2/kWh and US$0.3/kWh.

1. Devices

bASIC's 54Ghps model energy consumption has been updated to 125W.

2. The charts

Chart 1: (new) In order to read this group of charts, find the intersection of a percentage ROI and number of difficulty periods (eg. % ROI after one year is at ~ 26 difficulty periods). The colour of the tile is an indicator of the exchange rate required to meet this %ROI after the given number of difficulty periods. The faint white line along the middle of each plot indicates the break even point. Click on a chart for enlargement.

Chart 2:  (Updated) Profitable mining limits as a function of exchange rate and mining Difficulty.

The chart below shows the profitable mining limit for four ASIC devices for which I was able to obtain hashrate and power consumption estimates. It can be used to find the maximum difficulty at which an ASIC device can profitably mine. For example, at an electricity cost of USD$0.20 per kWh and an exchange rate of USD$20 / btc, the BFL Single SC plus a 100 W computer will be still profitable at a difficulty of 750 million, but not at a difficulty of 1 billion.

Chart 3:  (Updated) Profitable mining duration as a function of exchange rate and time since purchase.

The chart below shows a comparison between the four devices listed (an additional 100 W added for the attached computer). The chart facets are groups according to assumed D when mining on the device started and the electricity cost. Pick a column closest to your local electricity cost, a row closest you the D at which you think you'll be mining, then if you agree that the average percentage increase in D per difficulty period is 5% you can determine for how long the devices are likely to be profitable.

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