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9 100 miners 200 miners 300 miners The time when a runner-up finds a valid block, during the first round, can be estimated as follow: t = − 1 log (1 − p) (13) λ(1 − hprev ) where hprev is the sum of the hashing power of all its predecessor runners-up including the first winner of the round. The time t is increasing for every newer runner-up as the ratio of the total network power is decreasing (1−hprev). We conduct extensive simulation and average the power saving in Green-PoW over 100,000 blocks. We consider three network sizes with 100, 200, and 300 miners, and different hashing power using Uniform and Normal distri- bution. Table 1 summarises the simulation parameters used to assess power saving. TABLE 1: Simulation Parameters 6.3 Results Fig. 5 and Fig. 6 show the impact of the number of runners- up and the size of the network on the total energy con- sumption in Green-PoW. Fig. 5 illustrates the ratio of power saving in Green-PoW with respect to the original PoW when varying the number of second round contenders for different network sizes. When only one node mines the block in the second round, the saving power is nearly 50% regardless of the size of the network. However, for a larger number of winners, the saving drop to nearly 32%, 41%, and 44% for networks of 100, 200, and 300 nodes, respectively. We also evaluate the total energy consumption in PoW and Green-PoW during the first and the second round and plot the results in Fig. 6. For a network of 100 miners, as shown in the figure, in Green-PoW the energy consumption during the second round is nearly 10% of that of the first round; such dramatic energy saving is due to the fact that only few nodes are participating in the mining process during the second round. In PoW, the average energy consumption is almost constant and is 8-10 times more than the second round of Green-PoW. Green-PoW consumes more energy than PoW in the first round since the nodes continue min- ing the same block in order to determine the runners-up. Nonetheless, the average of the first and second rounds is about 30-50% less than PoW. We also assess in Fig. 7 the impact of a different distri- bution of the hashing power on the energy-saving in Green- PoW. We consider a network of 200 miners and engage 5 nodes to mine in ρ2. We distribute the hashing power among network miners by varying the percentage of miners that hold 50% of the total network hash power, while assuming the remaining power is equally distributed among the other 50% of the network. When 50% of the network, i.e., 100 miners, equally hold 50% of the hashing power, this means that all miners have exactly the same portion of hashing power (0.5%). As illustrated in the figure, when a small 50 45 40 35 30 Fig. 5: Energy saving ratio Vs. number of second round miners 15 10 5 0 Fig. 6: Normalized energy consumption Vs. number of second round miners portion of miners (2%) holds most of the hashing power (50%), the energy-saving in Green-PoW is minimal, how- ever, when the power is equally distributed among miners, Green-PoW achieves its maximal saving. This is mainly due to the fact that the energy-saving in Green-PoW depends on the mining power of the second round contenders. When some of them have high power, per equation (4) the mining difficulty will be increased and consequently, more energy needs to be spent in order to find the second round block. Vise versa, when they have small hashing power, less energy will be consumed in order to find the block in the second round as the mining difficulty will be reduced. 6.4 Time-out and η Selection As we discussed in Section 4, the time-out and η are two important parameters that help in striking a good balance between system liveness and energy-efficiency. The η pa- rameter defines the additional time a particular node needs to spend in mining during ρ1i after hearing from the first considered runner-up. This time can be set as a function of the number of miners that we want to have in the second 1 2 3 4 5 6 7 8 9 10 # second round miners Parameter Value # blocks 100, 000 # miners [100, 200, 300] # second winners [1, 2, ..., 10] Hashing power dist. % of miners having 50% of total hash power [2%, 5%, 10%, 20%, 50%] 1st round 2nd round PoW 1 2 3 4 5 6 7 8 9 10 # second round miners Normalized energy consumption. (E/P) Energy saving (%)PDF Image | Efficient Blockchain Proof-of-Work Consensus Algorithm
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