OptimalBlockDelayTimeToBlockTimeRatio

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By knowledge mentioned in EstimateBlockFee and BlockIntervalTime, one can derive the optimal ratio about the delayed time of block propagation. When miners assess zero revenue for orphan probability and aims to maximize the the expected value of the block fee, they come to maximize

With x being defined such x = B / MT = ND / M and k being defined k = M / DnA , this formula is translated to

which is maximized at x when

When k is large, the optimal x approaches zero. As mentioned in a long run the n or A may adapt with the x decided by the miners to have long term kx being 1 and then the expected value becomes

Precisely speaking, the orphan does not necessarily mean zero revenue because the first chain might lose to the appeared orphan or not. With knowledge mentioned in ScaleDebate, the overall impact is the reduced production efficiency and the miners are to maximize which, being decreasing beyond 1/k, is maximized at x when or if block reward R is considered, and must be between 0 and 1/k.

Two calculation examples.

  • M = 5 mega byte, D = 266 byte, nA = 1000, a miner with h = 10%, then k = 19.71, the optimal x is 0.0486, meaning the tx queue ratio N / nA being 95.79%, the optimal block size being 145.8 mega byte, the loss of production efficiency due to orphan/reorg being 4.17%.
  • M = 5 mega byte, D = 266 byte, nA = 100000, a miner with h = 10%, then k = 0.1971, the optimal x is 1.9, meaning the tx queue ratio N / nA being 37.45%, the optimal block size being 5700 mega byte, the loss of production efficiency due to orphan/reorg being 58.97%.

Note that the two maximization agree on the first order by the Taylor series expansion at x = 0 when the miners are small meaning h being almost zero. Again, in a long run the n or A may adapt with the x decided by the miners to have long term kx being 1 and then the expected value becomes . By the point of view of the macro, it is the reduced price of the coin and a regular block fee of qVT. As time goes by, k increases because miners choose higher M for the sake of higher block fee without sacrifice the security as mentioned in AttackCost.