# bitcoin block time analysis with conjuction to propagation

For both scenarios that I will describe below, assume the 2 things.

• that network propagation for blocks always 6 seconds.

• that we start looking at my analysis from time = t0 and at that t0, every node has the same exact chain.

Scenario 1: block time = 10 min

If block time is 10 min and minerA solves blockD at time t0 + T, and shares it immediatelly, during the time t0+T and t0+T+6, others miners might solve their own block that they were working on. Let's call the variable how many miners solve it during this time to be "X"

Scenario 2: block time = 4 min

If block time is 4 min and minerA solves blockD at time t0 + T, and shares it immediatelly, during the time t0+T and t0+T+6, others miners might solve their own block that they were working on. Let's call the variable how many miners solve it during this time to be "Y"

As it turns out, Y > X.

Question 1: Am I right that Y would be greater than X ? ofc, not in 100% cases, but in terms of probability.

Question 2: How do I make myself sure that it's mathematically true that Y > X ? I know it must be about how difficulties and target are set. It seems like the less difficulty, The higher the chance to solve a block during ANY 6 second period. (NOTE the word: "ANY"). This is important because we don't know exactly when minerA solves the block, but as I've read, probability that Y > X is true for ANY 6 second period, and this 6 second period doesn't have to be closer to 4 minutes or 10 minutes or whatever it is. What would be a mathematical approach to this so I believe in this ?

Am I right that Y would be greater than X [...] in terms of probability.

## block time := actual interval between recent blocks

Not if by "block time" you refer to a recent interval between the datestamps on two successive blocks. Or between the arrival times at one node of two successive blocks.

It is entirely possible that three successive blocks are produced with successive intervals of 10 and 4 minutes.

The probability of any miner succeeding in producing a hash less than the network target in any small interval, like 6s, is proportional to their hashpower. It is also inversely proportional to the current network difficulty. It is not affected by recent intervals between block publication.

I think the maths supporting this can be found by using this sites search facility and looking for "poisson"

## block time := target for average interval between blocks

To make the average of intervals shorter, the network would have to set a lower difficulty. At the least, this would be the simplest and most obvious way to cause that result so far as I know.

As noted above, probability of a hash being less than a target is inversely proportional to the difficulty. So the probability would be higher.

This sort of change alters the ROI of miners and would affect the number of miners and the amounts they were willing to periodically invest in new or additional equipment. I think the dynamics of this might mean that a miner who doesn't invest in extra equipment might not see the expected change in success rates endure for long.

• by block time, I mean, the bitcoin's 10 minutes block time that in 2008, satoshi decided. I think that block time is averagely 10 min because target gets adjusted such as it takes averagely 10 min to solve the block. and i bring 2 scenarios, what if target were adjusted always to cause 4 min block time averagely instead of 10. then read the question again and In this definition of block time, does anything in your answer change ? Commented Feb 23 at 15:20
• @Giorgi: Yes it does change my answer. See above. Commented Feb 23 at 15:37
• ah, so I was right claiming that Y > X(I meant block time as in your 2nd approach) . correct ? Commented Feb 23 at 15:37