Author: Tripoli
Original compilation: 0x11, Foresight News
Author: TripoliReddit AMAOriginal compilation: 0x11, Foresight News
Earlier this week, I was at the Ethereum Foundation
A question was asked on , and one of the responses caught my attention:
- /u/vbuterin
I'm looking forward to seeing how transaction packaging times change after the merge!
Theoretically, the average packing time after the merge should be reduced by about one-half, because the expected time for the next block is now 6 seconds, compared to 13 seconds before, and the more regular packing time also reduces the peak. In my personal experience, transactions are packaged very quickly today, even compared to the period after EIP-1559, before the merger. It would be interesting to see the data in detail.
Bitcoin block time dynamics are more studied and data available. Therefore, I will transfer Ethereum's proof-of-work to Bitcoin for analysis [1]. The two systems are very similar, with the main difference being that the difficulty adjustment keeps the average block time at 10 minutes for Bitcoin and around 13 seconds for Ethereum [2].
However, blocktime averages are a huge simplification. The media often refers to Bitcoin miners as a group of computers solving complex mathematical problems, but this is not the case. What the word solve implies is that mainstream analogies like these show mining rigs getting closer to a solution, but that's not how cryptomining works. The simplest analogy for hashing is to toss a coin and try to get heads 77 times in a row [3]. Incorrect hashes do not provide meaningful progress, which is a version of the gambler's fallacy: all hashes are independent, and there is no progress when a hash fails.The attempted independence and resulting lack of progress has been described as amnesia. Regardless of how much time has passed since the previous block, the probability that a Bitcoin miner will find the next block is always the same: the average time the next block will appear is always 10 minutes away.great article
. In simple terms, block intervals follow an exponential probability distribution, ignoring edge cases originating from very rapidly occurring blocks.
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In order to discuss transaction packaging time, we need to add another layer to the analysis. Naively speaking, when a Bitcoin transaction is committed, it should be included in the next block, which will be completed in 10 minutes on average; therefore, the block time should be 10 minutes. In practice, the competitive nature of block space distorts block times and challenges assumptions about exchangeability.
The graph below shows the climb in pending transactions on September 2, 2022. Each cliff or drop in the graph represents the discovery of a new block and the processing of a batch of the most valuable transactions. For convenience, the grid vertical lines are at intervals of 10 minutes, which is the average block time of Bitcoin.
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Bitcoin mempool
The graph shows an atypical but not uncommon gap in block discovery. From 10:20 to 12:18, only two blocks were found instead of the expected twelve.
While it seems safe to send a 12 sat/vB transaction at 10:20, it may not be processed until 13:00, at least 8, possibly as many as 11 blocks. Two sources of delay are block time variance and asymmetric time information, i.e. transactions submitted later have the advantage of additional information and are able to pay a larger fee to skip the transaction queue.
Taking a random sample of slowly discovered blocks and analyzing the transactions contained therein, a clear pattern emerges between transaction commit times and the relative density of transactions included in the next block [4]. Slow blocks contain approximately 50% more transactions from the time bin immediately before the block was discovered than from the time bin immediately after the previous block.
In sharp contrast to this, if we look at fast blocks, the distribution of commit times is much more even [5]. This shows that time competition in fast blocks does not make sense.
Most blocks are found pretty quickly, so does it matter?
If we calculate the expected block time, the contribution of long tail blocks is actually surprisingly large. If one calculates the average block time but ignores outliers like 20+ or 30+ minutes, the expected block time drops to 6-8 minutes.
Regarding packing time, the time competition for block space basically ignores these long tail blocks. For example, if we assume that transactions older than 20 minutes are replaced by new higher-fee transactions, the average inclusion time for transactions entering the first block should theoretically drop to only 6 minutes.
Actually, it didn't go down to 6 minutes. We're seeing transactions not being fully replaced in the slow block distribution, but I'd expect the first block time to be probably in the 8-9 minute range, rather than the naive 10 minute approximation.
Further counting of replaced transactions would require a more comprehensive analysis of the data than presented in this paper (heavily dependent on trends in block space requirements, subsequent random rollover of block intervals, etc.), and the result could be a multimodal harmonic distribution.
Returning to Proof-of-Stake Ethereum, assuming non-competitive block space, a constant 12-second block interval suggests an average block time of 6 seconds [6] (compared to 13 seconds in Proof-of-Work).
However, competition for Ethereum block space is fierce.
Even with constant block intervals, we seem to be seeing intense competition for time, even more than competition for Bitcoin long-tail block space. All of this happens within seconds, repeating every 12 seconds.
So, is the packing time shorter? I guess it will depend on how pack time is defined. If the first block on Ethereum is dominated by transactions submitted in the last few seconds of the block interval, then it's not clear to me that the drop in variance would make a meaningful difference. On the other hand, if we are talking about a single transaction whose priority fee is sufficient to ensure that the transaction is included, then the expected packaging time should be reduced from 13 seconds to 6 seconds [7].
annotation:
[1] The variance of the exponential probability distribution is proportional to the square of the block time, so the variance of the Bitcoin block time is ( 600/13)^ 2 = 2130 times that of the Ethereum PoW block. Perhaps this explains the relative lack of analysis of Ethereum block times.
[2] This indicator assumes that Ethereum is not in the time window of the hard fork, in which the block time begins to increase exponentially to force miners to adopt this change.
[3] As of this writing, the 30-day average hash rate is 250.75 million TH/s, so there are 250.75 × E 6 × E 12 × ( 60 × 10) = 1.5 E 23 times in a 10-minute block Hash try. To find a coin flip equivalent, we take a base 2 logarithm that is equivalent to 76.9 consecutive coin flips.
[4] The slow blocks in this sample are pseudo-random sequences found between December 22, 2022, and January 12, 2023, containing 40 blocks with block intervals between 42 and 75 minutes .
[7] Unfortunately, the pre-merge data does not appear to be reliable, and while this could just be due to the hyper-competitive nature of block space coupled with block time differences, I'm not confident enough to publish it.
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