## 3 Bitcoin Mining Activity: The Model
### Total Power Consumption in a Hypothetical Bitcoin System Using PoS  
In the following, we computed the total power consumption in a hypothetical Bitcoin system using PoS as a consensus mechanism. This mechanism is implemented as in the Nxtsystem, which is a 100% PoS cryptocurrency system that is less popular than Bitcoin (see [29, 30]).  
In the real Bitcoin network, the miners have to run their mining hardware continually in order to secure the network. Contrary to Bitcoin, in the Nxt system, everyone who owns Nxt can be chosen to protect the network. The probability to be chosen is proportional to the Nxt owned. With this mechanism, the computers run to validate the transactions and not to secure the network.  
Figure 4: Average and standard deviation (**a**) of the power consumption expressed in Watts to power the mining hardware units and (**b**) of the total expenses expressed in dollars, including the expenses needed to set up and maintain the mines, across all Monte Carlo simulations in the market using PoW.  
Figure 5: Average and standard deviation of the total mining cost per mined Bitcoin expressed in dollars across all Monte Carlo simulations.  
In general, in a cryptocurrency market using PoS, everyone holding an amount \(b_{i}(t)\) of Bitcoins has a probability higher than zero of mining Bitcoins. This probability is proportional to the cryptocurrency owned by the \(i^{\text{th}}\) user, \(b_{i}\)[30].  
Everyone holding Bitcoins can be a potential miner, but replicating the work by [PERSON] [31], we hypothesized that the validation of a block involves a number of miners (In the Nxt system, people creating blocks are called \"forgers\". This stems from the name of the process of block generation known as \"forging\".) equal to three, which are in the full power state while forging (in [31], the author assumed three forgers and not only on,e because multiple forgers, operating at the same time, keep each other honest and increase the network security).  
Note that miners do not belong to a pool. This is because, in a system based on PoS, there is no arms race for acquiring specialized hardware needed to run computations, and hence, there is no need to pool together to share resources. PoS is CPU friendly; consequently, we assumed that each miner owns a machine characterized by a power consumption equal to 130 W while she/he mines.  
The power consumption in the hypothetical system is computed simply taking into account the power consumption of the mining machines. Assuming that the potentialities of the mining hardware do not vary in the simulation period and fixing the number of machines involved in the mining activity of a block equal to three, the machine power consumption per hour equals 130 W; the time needed to validate a block equals ten minutes; and the number of blocks per day is equal to 144; it follows that the power consumption per day is constant and equal to:  
\[\frac{130\ W}{6}\times 3\times 144=9360\ W.\]  
Thus, the cost per mined Bitcoin is equal to \(\$0.442\times 10^{-3}\), the number of Bitcoin per block being equal to 12.5. Of course, the cost per mined Bitcoin is lower in this PoS system than that estimated for the system using PoW.