234078 Longest block to solve yet?

[flatfly]

Just curious, what is the longest time that a block has stayed unsolved?
For instance, 234078 took 1 hour and 2 minutes to solve.

[Come-from-Beyond]

Just curious, what is the longest time that a block has stayed unsolved?
For instance, 234078 took 1 hour and 2 minutes to solve.

I recall being waiting for 97 minutes.

[deepceleron]

Most long blocks came early in Bitcoin, here's 212 blocks (over 100 minutes since last block timestamp)

Note that the block timestamp is based on the mining computer's time, there are more blocks than this with a negative time.

15324 1508.9  
16564 1506.5  
15     1452.6  
16592 1229.7  
20189 1003.4  
19724 785.5    
21438 637.0    
19722 625.6    
28     514.7    
16490 511.4    
20432 507.1    
20364 500.2    
20349 488.8    
21389 468.7    
19565 433.3    
15048 422.9    
19726 415.1    
74638 411.3    
21446 392.9    
21527 384.7    
21466 384.2    
16468 380.7    
26242 368.3    
21359 357.8    
21449 342.3    
21463 330.1    
21382 327.1    
21376 321.3    
21383 308.6    
23418 305.4    
21585 302.0    
23434 300.7    
20384 299.6    
16215 294.8    
15424 285.1    
23433 277.6    
21442 263.8    
16286 263.8    
21464 261.6    
22945 260.1    
22527 259.3    
20325 248.2    
21458 238.0    
21374 234.9    
19725 234.2    
21418 232.2    
21444 230.8    
28705 226.3    
21440 225.9    
21348 225.1    
1390   224.3    
11964 223.8    
169   222.4    
21462 220.2    
21403 217.6    
21331 217.3    
20435 216.1    
8231   214.7    
20439 211.4    
1398   211.2    
32647 208.9    
13889 208.9    
20190 205.3    
16271 202.3    
20187 198.4    
21361 196.8    
13898 196.4    
1917   196.1    
21340 195.0    
25788 194.1    
26237 191.6    
20405 190.9    
8211   190.7    
20357 190.7    
1296   188.3    
21385 188.2    
11966 186.9    
32629 186.3    
15228 186.1    
21393 179.2    
21459 178.8    
20344 175.6    
8226   175.2    
20008 174.9    
163   174.2    
21428 172.7    
20361 170.8    
20343 170.4    
18030 166.6    
79     165.9    
20339 163.7    
23425 163.0    
21380 162.4    
20436 161.9    
21584 161.5    
20418 160.8    
21345 160.1    
25740 158.1    
20417 157.4    
20351 157.3    
21581 156.5    
21443 155.1    
1916   155.1    
20308 155.0    
28742 154.3    
16588 154.0    
21447 150.8    
20427 150.4    
23426 148.7    
22950 147.5    
13082 146.9    
21441 146.5    
21332 146.5    
15331 146.3    
20390 145.0    
155290 144.8    
1389   143.5    
149098 141.0    
20310 140.0    
20356 137.2    
70718 135.9    
21423 134.5    
70665 133.2    
20304 132.8    
15818 132.8    
21369 132.2    
20333 132.0    
20376 129.3    
20188 128.7    
17149 127.9    
21325 127.5    
20438 126.2    
20421 124.4    
22018 123.0    
32527 122.1    
25132 122.0    
15222 121.8    
69515 121.4    
20203 121.3    
76594 121.2    
26275 121.0    
20191 120.9    
26236 120.6    
105909 120.5    
25889 120.3    
28702 120.2    
23421 120.1    
156113 119.6    
154185 119.4    
19951 117.8    
20411 116.4    
21586 116.2    
21360 115.9    
16590 115.6    
24419 115.4    
16565 115.3    
20359 115.3    
26843 114.6    
24956 114.4    
163966 114.0    
21323 113.4    
26873 113.2    
24415 112.8    
24494 112.7    
21422 112.5    
23430 112.3    
207497 112.2    
26249 112.1    
26201 112.0    
26190 111.6    
87107 111.2    
175720 110.5    
26212 110.1    
20327 109.6    
21451 109.5    
18253 109.4    
28731 108.2    
21334 108.0    
26104 107.5    
8223   107.0    
205771 106.5    
22158 106.3    
21461 106.2    
32632 105.9    
13892 105.7    
21433 105.7    
25130 105.5    
206669 105.4    
28697 105.3    
152996 104.7    
204798 104.4    
20347 104.3    
18265 104.3    
25033 104.3    
20334 104.2    
20381 103.8    
21408 103.5    
28771 103.5    
8227   103.1    
26260 102.9    
21597 102.3    
154346 102.2    
28749 102.1    
19977 102.0    
25117 101.9    
24263 101.8    
21373 101.5    
26126 101.3    
15230 101.0    
25147 100.2    
26251 100.1  

[dooglus]

As global hashrate increases, difficulty is increased to keep the mean time between blocks at 10 minutes.

But what happens to the variance of the time between blocks as the difficulty increases?  From the above stats it would appear that it decreases.

[jackjack]

Yeah it decreases

[FreeMoney]

Can someone explain how that works? The variance decreasing?

[Zeilap]

My stats are a little rusty, but as far as I see, it doesn't, it goes up.

Hashing is Bernoulli trial, either you beat the target or you don't. That gives mining a geometric distribution, and so the variance (in number of hashes required) is (1-p)/p². The re-targeting algorithm tries to keep the hashrate directly proportional to 1/p, so the variance in time quadratically increases with the hashrate. Right?

I guess the phenomenon above was a much higher variance in hashrate because of people turning their computers off at night, whereas now the variance is greatly reduced due to increased numbers and because miners tend to keep the gear running 24/7.

[flatfly]

nice list, deepceleron, thanks

[deepceleron]

My stats are a little rusty, but as far as I see, it doesn't, it goes up.

Hashing is Bernoulli trial, either you beat the target or you don't. That gives mining a geometric distribution, and so the variance (in number of hashes required) is (1-p)/p². The re-targeting algorithm tries to keep the hashrate directly proportional to 1/p, so the variance in time quadratically increases with the hashrate. Right?

I guess the phenomenon above was a much higher variance in hashrate because of people turning their computers off at night, whereas now the variance is greatly reduced due to increased numbers and because miners tend to keep the gear running 24/7.

Only 'technically', on a very small scale, is the variance different. In reality it is unobservable at different difficulties.

Hashing is typically modeled as an exponential distribution, a continuous series, however it is actually a geometric distribution. The probability is so low though, that the R statistics package fails to compute due to overflow errors with statistics on much more than difficulty 1.

The variance is given for a geometric distribution as:



You can see that for extremely small p (probability of one hash finding a block), the top numerator approaches 1, becoming insignificant compared to the p².

The high block times around 20000-30000 has something to do with the satoshi miner and others not making much hashes during that time, there was much less than difficulty 1 worth of mining being done. I could eliminate the first year of Bitcoin to get a better list of long blocks, long due to variance rather than low hashrate.

[Zeilap]

Only 'technically', on a very small scale, is the variance different. In reality it is unobservable at different difficulties.

I was hoping you'd explain why the difference is unobservable.

Anyway, if anyone has the time to do it, it might be interesting to see a chart of the cumulative distributions of the solution times during e.g. January and April, superimposed to see if the bell shape has noticeably widened with the increase in hashing power.

[deepceleron]

It's not a bell curve, the probability density function of an exponential distribution looks like this:



Note that the average block time is 10 minutes (1/λ), but 50% of the blocks are less than 6.93 minutes (the median ln(2)/λ).

The reason that it makes no difference is: whether the geometric distribution (which counts individual hashes) is calculated for difficulty 1 or difficulty 10000000, the probability of a quantile (such as calculating the probability of a block taking 100 minutes) is the same within several decimal points. In fact it is already identical to the exponential (continuous) function with three digits by difficulty 1, higher difficulty just makes the density function of geometric converge to exponential with even more digits of identity.

http://math.stackexchange.com/questions/93098/how-does-a-geometric-distribution-converge-to-an-exponential-distribution

Here's a monte carlo simulation of a geometric distribution density, the red line is exponential:

See how much like the exponential it is? This example shows a 0.1 probability; Bitcoin's current probability is 0.0000000000000000231. With a lower probability,the steps disappear and the distribution converges to exponential.

So the answer is: the probability of a block taking 62 minutes (given a constant hashrate equivalent to difficulty) is 0.2029%, whether the difficulty is 1 or a billion.

[Zeilap]

Deepceleron, thanks for taking the time to explain, though the convergence of the geometric to exponential distribution wasn't the issue I had. However, dealing with the exponential distribution made finding my error much simpler.

My issue was the conversion from dealing with number of hashes to dealing with time.

The re-targeting algorithm tries to keep the hashrate directly proportional to 1/p, so the variance in time quadratically increases with the hashrate. Right?

My incorrect reasoning was that 1/p is the mean number of hashes, which we choose to be 10 minutes worth of hashes at the current hashrate, R, so  1/p = 600R, then 1/p² is 600²R² and concluding that this is the variance in time, which therefore goes up with the square of the hashrate .

Anyway, I worked out the error: Transforming the exponential CDF for the number of hashes required into the CDF of time required using  1/p = 600R  gives an exponential distribution with a mean of 10 minutes, in particular the hashrate cancels itself out, and so the variance is constant, no matter the hashrate.