3.1 The rise and fall of Arsbitcoin.com

0. Introduction


Pool: Arsbitcoin.com
Pool op: Burning Toad
Publicly identifiable pool operator: No
Payout method: Shared Maximum Pay-Per-Share (SMPPS)
Current pool hashrate: 150 Ghps
Availability of public statistics: All rounds available but not formatted
Accuracy of public statistics: Two obvious errors, one in round time (round 445) and one of round length(round 520). Round duration for first round unavailable. Another reported error: if two blocks were mined in quick succession, one would not show up in the pool stats.
Are public statistics reflecting actual data? Probably not (see below)
Public perception of pool and pool op: Generally good.


Should I mine here? No


Arsbitcoin.com bitcoin mining pool found their first block on 24th June 2011 at about 50 Ghps.  By August 2011 Arsbitcoin.com's hashrate had increased to 400 Ghps, and peaked at 800 to 900 Ghps from late September to early November 2011. From this point, amid complaints from some miners of the pool dropping up to 25% of submitted shares and servers down and no response from the pool operator, the hashrate entered a much quicker decline to 300 Ghps by early December 2011. 


At this point Burning Toad made a statement to the effect that due to a situation beyond his control he would be unable to manage the pool as closely as before and may need to close it. The hashrate then remained stable until early February 2012 at which point the published SMPPS buffer was more than -1200 btc and payouts were significantly delayed. The hashrate decreased again to its current low of 150 Ghps. What caused such a sudden change in the pool's fortunes? Is it safe to mine there?

1. Do Arsbitcoin.com round lengths appear geometrically distributed?
Note: Data used in this post is available here as a .csv file. Each round is Unix time timestamped, and round lengths are divided by the Bitcoin difficulty (D) at the time.


As mentioned in previous posts, pooled mining round lengths should be geometrically distributed. Based on public perception of the pool's trustworthiness, I expected to see no unusual results. However, that is not what we see here.


Below are Arsbitcoin.com's round lengths as a fraction of D at the time the block was solved, as a histogram, and as boxplots, all compared with simulated rounds. Note that Arsbitcoin round 520 (block number 147820) was removed (0 shares/round recorded).

Looking carefully at the first comparison, we see that Arsbitcoin.com's round lengths are all larger than 0.01xD, and for most of the time period none under 0.02xD. At current D, that's about 15000 - 30000 shares as minimum round length. Further, the cutoff appears consistent and artificial.


The histograms look similar until we reduce the bin width at which point  the far fewer counts in the first bar are in clear contrast with the the second bar with three times the count. This means that shorter round lengths happen much less than usual, but slightly longer round lengths occur much more than usual.


Boxplots show the same thing as the round length comparison. Very few outliers, with no outliers below 0.01xD and few below 0.02xD, compared to either Deepbit.net or the simulated rounds.

2. Theoretical comparisons - statistical parameters.

Mean, median, variance, skewness and kurtosis are statistical parameters that we have previously used to analyse the distribution to which a pool's round length random variable belongs. The theoretical geometric distribution statistics are: 

mean   = 1/p = D
median   = -1/log(D,base=2) ~ 0.693*D
std.dev. = sqrt((1-p)/p^2) ~ D
skewness = (2-p)/sqrt(1-p) ~ 2
kurtosis = 5-p+1/(p-1) ~ 6

The statistical parameters of Arsbitcoin.com round lengths are:

mean  = 1.002211
median  = 0.6762516
st.dev  = 1.01667
skewness = 2.16698
kurtosis = 7.748342

Round lengths are already divided by D so the parameters of Arsbitcoin.com's round length distribution are quite close to the values expected for a geometric distribution, except for a much larger kurtosis (heavier tail), which may be related to the abnormal shorter round lengths.

 3. Theoretical comparisons - quantiles and empirical cumulative distribution function.
If you have read post 2.1 Deepbit.net, then you'll already know this but the following is worth repeating. 

The cumulative distribution function (CDF) of a Bitcoin pooled mining round lengths describes the probability that a round will be solved after a given number of shares have been contributed. For example, a large probability means that most rounds will be solved when a large number of shares are submitted. Since shorter rounds are more common, we expect the CDF to rise rapidly, and then level out and approach 1 at larger round lengths. The comparison plot below on the right shows very little difference between the theoretical CDF for geometrically distributed variables and the empirical CDF (eCDF) for Arsbitcoin.com round lengths. The ecdf is is defined as:

ecdf = n/max(n) where 'n' is the nth datapoint in a set of data ordered by size

The Kolmogorov-Smirnov test  can be used to compare an eCDF to a theoretical CDF.In this case:

If:

H0 = Arsbitcoin.com's round lengths belong to a geometric distribution,

HA = Arsbitcoin.com's round lengths do not belong to a geometric distribution, then:

p value = 0.9621

This means we cannot reject H0 as a hypothesis, and provides evidence that Arsbitcoin.com's round lengths are distributed geometrically. 

Next we use a QQ plot (quantile-quantile) to compare the quantile from Arsbitcoin.com's round length distribution and what that distribution should be theoretically. Quantiles are points taken at regular intervals from the cumulative distribution function (CDF). The empirical quantiles are from Arsbitcoin.com's round length data, and the theoretical quantiles are calculated using the number of rounds in the dataset, so that there are the same number of theoretical as empirical data, with the same cumulative probabilities (from 1/1663 and up). The empirical quantiles are then plotted as a function of theoretical quantiles, and the relationship here should be y = x. From the plot below left we see that the relationship seems to be approximately y=x. However, the QQ plot on the right covers the shortest 150 rounds and confirms that empirical round lengths for the first part of this range are larger than expected.

The cumulative distribution function or CDF describes the increasing probability of a particular round length solving a block. Below are the empirical and theoretical CDFs.

Here we see that on the largest scale, the CDFs are a good match. However, the shortest rounds have a much lower cumulative probability than they should.

4. Is Arsbitcoin's mean round length probable?

In a word, yes. Arsbitcoin.com's mean round length is 1.002211xD, very close to the expected mean of D. However, what seems 'common sense' is not always the case. so we'll complete the analysis.

To do this we use the Central Limit Theorem (CLT) as described in the Deepbit.net Neighbourhood Pool Watch post. As a reminder, the CLT states that a sufficiently large number of independent random variables will be approximately normally distributed, so we can calculate the probability using a normal distribution. This means we can calculate the likelihood of a mean round length if we also know the standard deviation of the the round lengths and the number of rounds:

population mean = sample mean
population std. dev. (sd) = sample sd / sqrt(number of variables in sample, n)

For Arsbitcoin.com:

population mean = D
sample mean = 1.002211
sample sd   = 1.01667
n = 1662 (with round 520 removed)
population sd = 1.002211/sqrt(1662) = 0.0245835

Using the normal CDF function in R, we get:

> pnorm(1.002211,1,0.0245835) 

[1] 0.5358319

This is definitely in the 'probable range as the plot below shows:

It should be kept in mind, however, that since it appears that block lengths are not recorded correctly on Arsbitcoin.com's published statistics, this may not be based on the real mean and standard deviation of round lengths.

5. So why shouldn't I mine at Arsbitcoin.com?

Apart from the problems mentioned in the introduction and the statistical anomalies described above, mining at a pool where the well meaning pool op is otherwise occupied and also has an SMPPS payout system with a large buffer and a declining hashrate is likely to leave you out of pocket - either for sometime until the buffer increases to positive, or forever if the pool fails. I'll describe the problems with SMPPS as it pertains to Arsbitcoin.com in the next post.

I don't think that the abnormal round lengths are due to malice - I can't see how an SMPPS pool would profit from them unless shorter rounds are being stolen by the pool op (which can't be discounted). It is more likely a failing of the pool's software. However, it should be seen as another warning sign.

6. Conclusions.

• Public perception of the pool operator and the pool is positive and miners there seem to trust the pool operator.
• The pool operator is not publicly identifiable.
• The pool has been very popular in the past with few complaints until problems began to occur in November 2011.
• Arsbitcoin.com has a large negative buffer and a falling hashrate - two very big problems for an SMPPS pool.
• The pool op, Burning Toad, is currently unable to keep the pool running smoothly, as evidenced by the Arsbitcoin.com thread on bitcointalk.org and the round length anomalies described in this post.
• Next post will cover the problems with the SMPPS system.

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