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# A stochastic model and social optimization of a blockchain system based on a general limited batch service queue

• * Corresponding author: Shunfu Jin
• Blockchain is well known as a database technology supporting digital currencies such as Bitcoin, Ether and Ripple. For the purpose of maximizing the overall revenue of the blockchain system, we propose a pricing policy to impose on transactions. Regarding the mining process as a vacation, and the block-verification process as a service, we establish a type of non-exhaustive queueing model with a limited batch service and a possible zero-transaction service. By selecting the beginning instant of a block-verification process as a Markov point and using the method of a generating function, we obtain the stationary probability distribution for the number of transactions in the system at the Markov points and analyze the elapsed time for the mining cycle. Based on the model analysis results, we derive the average latency of transactions and demonstrate how the average latency of transactions changes in relation to the arrival rate of transactions. With a reward-cost structure, we construct an individual benefit function and a social benefit function. By improving the Grasshopper Optimization Algorithm (GOA), we search for the Nash equilibrium and the socially optimal arrival rates of transactions. Numerical results show that the Nash equilibrium arrival rate of transactions is always higher than the socially optimal arrival rate of transactions for a given mining parameter and a specific block capacity. For this, we propose a pricing policy that forces the transactions to accept the socially optimal arrival rate and maximize the overall revenue of the blockchain system, including all transactions and miners.

Mathematics Subject Classification: Primary: 60K15, 60K25; Secondary: 68M14.

 Citation:

• Figure 1.  The working flow of a mining cycle

Figure 2.  Operation of the queueing model with general limited batch service

Figure 3.  Change trend of the individual benefit $U_{I}(\lambda)$ of a transaction

Figure 4.  Change trend of social benefit $U_{S}(\lambda)$ of transactions

Figure 5.  The working flow of the enhanced GOA

Figure 6.  Nash equilibrium and socially optimal arrival rates of transactions

Table 1.  Numerical results for the remittance fee

 Mining parameter $(\theta)$ Block capacity $(b)$ Socially optimal arrival rate $(\lambda^{*})$ Maximum social benefit $(U_{S}({\lambda^{*}}))$ Remittance fee $(f)$ 0.5 40 11.0493 101.0826 9.0397 0.5 80 22.0986 202.2892 9.0996 1.0 40 20.9083 225.1180 10.6713 1.0 80 41.8166 450.5689 10.7271 1.5 40 28.0302 318.0631 11.2554 1.5 80 56.2193 636.5389 11.2767
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