2. Network propagation: Assume a simple model for how quickly Bitcoin blocks propagate through the network after t seconds, a group of miners controlling a proportion (t) of the mining power has heard about the transaction, up to some point tch all miners will have heard about it. That is, (t)-1 for all t>tmax . Further assume that (0)-0 and (t) is monotonically increasing in t. Assume that blocks are found in Poisson process with a mean time to find a block of B-600 seconds (1-1/600). A stale block (likely to become an orphan block) occurs if some miner finds a block before news of the most recent block reaches them a. Given the design of the Bitcoin P2P network, explain why an exponential propagation model (i.e. a(t)ocb' for some b) is a plausible model Hint: Recall that the derivative of an exponential is another exponential function. b. Suppose a(t)-2s0-1, that is, an exponentially increasing proportion of the mining power hears about a new block up until tax 30 seconds, at which point all have heard. What is the probability of at least one stale block being found? Note: You do not need to solve the integral analytically. You may use a computer. If we lowered to 60 seconds to make transactions post faster, how would this affect your answer from part (b)? What problems might this cause? One could argue that the increased rate of stale blocks identified in part (c) isn't really a problem as miners will still be paid at the same rate. Explain why this argument may not hold in practice. In particular, explain why the model for (t) from part(b) is incomplete. C. d. 2. Network propagation: Assume a simple model for how quickly Bitcoin blocks propagate through the network after t seconds, a group of miners controlling a proportion (t) of the mining power has heard about the transaction, up to some point tch all miners will have heard about it. That is, (t)-1 for all t>tmax . Further assume that (0)-0 and (t) is monotonically increasing in t. Assume that blocks are found in Poisson process with a mean time to find a block of B-600 seconds (1-1/600). A stale block (likely to become an orphan block) occurs if some miner finds a block before news of the most recent block reaches them a. Given the design of the Bitcoin P2P network, explain why an exponential propagation model (i.e. a(t)ocb' for some b) is a plausible model Hint: Recall that the derivative of an exponential is another exponential function. b. Suppose a(t)-2s0-1, that is, an exponentially increasing proportion of the mining power hears about a new block up until tax 30 seconds, at which point all have heard. What is the probability of at least one stale block being found? Note: You do not need to solve the integral analytically. You may use a computer. If we lowered to 60 seconds to make transactions post faster, how would this affect your answer from part (b)? What problems might this cause? One could argue that the increased rate of stale blocks identified in part (c) isn't really a problem as miners will still be paid at the same rate. Explain why this argument may not hold in practice. In particular, explain why the model for (t) from part(b) is incomplete. C. d