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8 PoW Green-PoW 30 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829 Miner ID Fig. 4: Shares of miners in Ethereum Vs. Green-PoW based imported from the main Ethereum network. 6 PERFORMANCE EVALUATION In this section, we evaluate the power consumption of Green-PoW using stochastic analysis. As modeled in the Bitcoin white paper [1] and in [25], [26], [27], the Bitcoin mining process could be well-approximated as a Poisson process with a deterministic rate λ which represents the mining rate or the average time between block-arrival events. In the following, we first formulate the mining process in both PoW and Green-PoW as a Poisson process and then assess the power saving achieved by Green-PoW. 6.1 Average Power Saving in Green-PoW Let M denote the set of n miners in the network M = {m1, m2, ..., mn}. Each miner mi ∈ M has a fraction hi of the total hashing power in the network P, so that it mines a new block at a rate of hiλ, where hi = 1. As explained in Section 3.1.2, the difficulty of finding a block is dynamically adjusted to ensure that a block is generated every 1/λ seconds in expectation with a rate λ (λ = 1/600 in Bitcoin). The inter-arrival times of consecutive blocks follow Exponential distribution with the same rate parameter λ, whose cumulative distribution function is: Pr[T ≤t]=1−e−λt (6) In PoW, each miner mi spends on average 1/λ and con- sumes energy Ei which is proportional to its hashing power hi and can be expressed as: Ei = 1 hi P (7) λ Therefore, the average total energy E consumed by the network to generate a block is inversely proportional to the block generation rate λ: Ethereum, calculated using real data of the latest 7438 blocks mining, additional energy is consumed in order to select the second-place winners. This additional energy depends on the number of second-place winners and the time they need to complete the mining and form a valid block. As- suming that mf is the first winner, ms, ms+1, ..., ms+k are k runners-up, and ts, ts+1, ..., ts+k the respective time needed by each of the runner-up to find the block. Thus the average total energy E1st consumed by the network during the first round can be expressed as follow: 1 k i=s j=i−1 k−1 E1st =Pλ+ti(1−hf + hj), (9) 1n 1 E=Pλ hi=Pλ (8) where hs−1 = 0 In the second round, the average consumed energy is pro- portional to the time needed to generate a block (1/λ) and the total hashing power of the runners-up. For simplicity, we do not consider the scenario where the set of runners-up fail to generate a block, and other miners start the mining process after the timeout: 1 k E2nd =Pλ hi (10) i=s From equations (8), (9) and (10), the power saving in Green- PoW can be, therefore, expressed as follow: Esave =2E−(E1st +E2nd) (11) 6.2 Experimental Setup In order to determine the time needed to select k runners- up and thus, calculate the energy spent in the first and the second round, we basically used the inverse function of the CDF in equation (6) and feed it different probability values p from a Uniform(0, 1) distribution to generate the blocks inter-arrival times t: 1 t = − λ log (1 − p) (12) i=1 In Green-PoW, a block is either generated during the first or the second round. In the first round, compared to PoW Mined blocks shares(%)PDF Image | Efficient Blockchain Proof-of-Work Consensus Algorithm
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