2 Network Propagation 23 points Lets sspumea simple model for how qucly blocks propagste through the network after tseconds, a group of miners controlligproportion at o the mining power has heard sbout the trancaction, up to some pointafter which all miners will have heard about it. Thst is, { t ) = 1 for all t z tme Further assume that a( o-o and { t ) always increases with increasing in t (ie. ( t ) is monotonically increasingin t) Assume that blocks are found in Poisson process with a mesn time to find a block of p-500 seconds A staie biock likely to become an orphan block occurs it some miner finds a block before news of the most recent s. Given the deaign of the Bitcoin P2P network, explsin why an exponential propsgation model (Le. a( t ) for some b ) s Psusible model. b Suppose a(tI-2*-i, thst is, an exponentielly increasing proportion of the mining power hears about a new block up until30 seconds, at which point l have heard Due to the memoryless nature of mining, the efmective amount of total time the entire sshrate of network spent lookingortle block is (2/s- sing this expression, what is the probsbility of at lesst one stale block being found Note: You do not need to solve the integral anaytica ly. You may use a computer your answer from part (b) (i.e. recalculate assuming P = 60 seconds)? What problems d. One could argue that the incressed rate of stale blocks identified in part cin't realy problem as miners willall sti be paid at the same rate proportional to their hashpower Explain why this argument may not hold in practice. In particular, explsin why our model for aft) from part (i incomplete. Explain how you might change the model to account for this. 2 Network Propagation 23 points Lets sspumea simple model for how qucly blocks propagste through the network after tseconds, a group of miners controlligproportion at o the mining power has heard sbout the trancaction, up to some pointafter which all miners will have heard about it. Thst is, { t ) = 1 for all t z tme Further assume that a( o-o and { t ) always increases with increasing in t (ie. ( t ) is monotonically increasingin t) Assume that blocks are found in Poisson process with a mesn time to find a block of p-500 seconds A staie biock likely to become an orphan block occurs it some miner finds a block before news of the most recent s. Given the deaign of the Bitcoin P2P network, explsin why an exponential propsgation model (Le. a( t ) for some b ) s Psusible model. b Suppose a(tI-2*-i, thst is, an exponentielly increasing proportion of the mining power hears about a new block up until30 seconds, at which point l have heard Due to the memoryless nature of mining, the efmective amount of total time the entire sshrate of network spent lookingortle block is (2/s- sing this expression, what is the probsbility of at lesst one stale block being found Note: You do not need to solve the integral anaytica ly. You may use a computer your answer from part (b) (i.e. recalculate assuming P = 60 seconds)? What problems d. One could argue that the incressed rate of stale blocks identified in part cin't realy problem as miners willall sti be paid at the same rate proportional to their hashpower Explain why this argument may not hold in practice. In particular, explsin why our model for aft) from part (i incomplete. Explain how you might change the model to account for this