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July  2021, 17(4): 1845-1861. doi: 10.3934/jimo.2020049

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

 1 School of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, China 2 Department of Intelligence and Informatics, Konan University, Kobe 658-8501, Japan

* Corresponding author: Shunfu Jin

Received  May 2019 Revised  October 2019 Published  July 2021 Early access  March 2020

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.

Citation: Wenjuan Zhao, Shunfu Jin, Wuyi Yue. A stochastic model and social optimization of a blockchain system based on a general limited batch service queue. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1845-1861. doi: 10.3934/jimo.2020049
##### References:
 [1] M. Aguiar and A. Lauve, Lagrange's theorem for Hopf monoids in species, Canadian Journal of Mathematics, 65 (2013), 241-265.  doi: 10.4153/CJM-2011-098-9. [2] A. Antonopoulos and O. Media, Mastering Bitcoin: Unlocking Digital Crypto-Currencies, O'Reilly Media, 2014. [3] R. Howell and E. Schrohe, Unpacking Rouché's Theorem,, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 27 (2017), 801-813.  doi: 10.1080/10511970.2016.1235646. [4] S. Kasahara and J. Kawahara, Effect on transaction-confirmation process, J. Ind. Manag. Optim., 15 (2019), 365-386. [5] Y. Kawase and S. Kasahara, Transaction-confirmation time for bitcoin: A queueing analytical approach to blockchain mechanism, 12th International Conference of Queueing Theory and Network Applications, LNCS, (2017), 75–88. doi: 10.1007/978-3-319-68520-5_5. [6] Q. Li, J. Ma and Y. Chang, Blockchain queueing theory, (2018). Available from: https://arXiv.org/abs/1808.01795. [7] R. Memon, J. Li and J. Ahmed, Simulation model for blockchain systems using queuing theory, Electronics, 8 (2019), 234-252.  doi: 10.3390/electronics8020234. [8] S. Nakamoto, Bitcoin: A peer-to-peer electronic cash system, (2008). Available from: https://www.coindesk.com/bitcoin-peer-to-peer-electronic-cash-system. [9] O. Novo, Blockchain meets IoT: An architecture for scalable access management in IoT, IEEE Internet of Things Journal, 5 (2018), 1184-1195.  doi: 10.1109/JIOT.2018.2812239. [10] W. Qian, Q. Shao, Y. Zhu, C. Jin and A. Zhou, Research problems and methods in blockchain and trusted data management, Journal of Software, 29 (2018), 150-159. [11] S. Ross, Stochastic Processes, Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1996. [12] S. Saremi, S. Mirjalili and A. Lewis, Grasshopper optimisation algorithm: Theory and application,, Advances in Engineering Software, 105 (2017), 30-47.  doi: 10.1016/j.advengsoft.2017.01.004. [13] M. Turkanović, M. Holbl, K. Kosic, M. Hericko and A. Kamisalic, Eductx: A blockchain-based higher education credit platform, IEEE Access, 6 (2018), 5112-5127. [14] M. Vlasiou, A non-increasing Lindley-type equation, Queueing Systems, 56 (2007), 41-52.  doi: 10.1007/s11134-007-9029-6. [15] L. Wang, X. Shen, J. Li, J. Shao and Y. Yang, Cryptographic primitives in blockchains, Journal of Network and Computer Applications, 127 (2019), 43-58.  doi: 10.1016/j.jnca.2018.11.003. [16] R. Wolff and Y. Yao, Little's law when the average waiting time is infinite, Queueing Systems, 76 (2014), 267-281.  doi: 10.1007/s11134-013-9364-8. [17] H. Zhao, P. Bai, Y. Peng and R. Xu, Efficient key management scheme for health blockchain, CAAI Transactions on Intelligence Technology, 3 (2018), 114-118.  doi: 10.1049/trit.2018.0014. [18] W. Zhao, S. Jin and W. Yue, Analysis of the average confirmation time of transactions in a blockchain system, 14th International Conference of Queueing Theory and Network Applications, LNCS, (2019), 379–388. doi: 10.1007/978-3-030-27181-7_23.

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##### References:
 [1] M. Aguiar and A. Lauve, Lagrange's theorem for Hopf monoids in species, Canadian Journal of Mathematics, 65 (2013), 241-265.  doi: 10.4153/CJM-2011-098-9. [2] A. Antonopoulos and O. Media, Mastering Bitcoin: Unlocking Digital Crypto-Currencies, O'Reilly Media, 2014. [3] R. Howell and E. Schrohe, Unpacking Rouché's Theorem,, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 27 (2017), 801-813.  doi: 10.1080/10511970.2016.1235646. [4] S. Kasahara and J. Kawahara, Effect on transaction-confirmation process, J. Ind. Manag. Optim., 15 (2019), 365-386. [5] Y. Kawase and S. Kasahara, Transaction-confirmation time for bitcoin: A queueing analytical approach to blockchain mechanism, 12th International Conference of Queueing Theory and Network Applications, LNCS, (2017), 75–88. doi: 10.1007/978-3-319-68520-5_5. [6] Q. Li, J. Ma and Y. Chang, Blockchain queueing theory, (2018). Available from: https://arXiv.org/abs/1808.01795. [7] R. Memon, J. Li and J. Ahmed, Simulation model for blockchain systems using queuing theory, Electronics, 8 (2019), 234-252.  doi: 10.3390/electronics8020234. [8] S. Nakamoto, Bitcoin: A peer-to-peer electronic cash system, (2008). Available from: https://www.coindesk.com/bitcoin-peer-to-peer-electronic-cash-system. [9] O. Novo, Blockchain meets IoT: An architecture for scalable access management in IoT, IEEE Internet of Things Journal, 5 (2018), 1184-1195.  doi: 10.1109/JIOT.2018.2812239. [10] W. Qian, Q. Shao, Y. Zhu, C. Jin and A. Zhou, Research problems and methods in blockchain and trusted data management, Journal of Software, 29 (2018), 150-159. [11] S. Ross, Stochastic Processes, Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1996. [12] S. Saremi, S. Mirjalili and A. Lewis, Grasshopper optimisation algorithm: Theory and application,, Advances in Engineering Software, 105 (2017), 30-47.  doi: 10.1016/j.advengsoft.2017.01.004. [13] M. Turkanović, M. Holbl, K. Kosic, M. Hericko and A. Kamisalic, Eductx: A blockchain-based higher education credit platform, IEEE Access, 6 (2018), 5112-5127. [14] M. Vlasiou, A non-increasing Lindley-type equation, Queueing Systems, 56 (2007), 41-52.  doi: 10.1007/s11134-007-9029-6. [15] L. Wang, X. Shen, J. Li, J. Shao and Y. Yang, Cryptographic primitives in blockchains, Journal of Network and Computer Applications, 127 (2019), 43-58.  doi: 10.1016/j.jnca.2018.11.003. [16] R. Wolff and Y. Yao, Little's law when the average waiting time is infinite, Queueing Systems, 76 (2014), 267-281.  doi: 10.1007/s11134-013-9364-8. [17] H. Zhao, P. Bai, Y. Peng and R. Xu, Efficient key management scheme for health blockchain, CAAI Transactions on Intelligence Technology, 3 (2018), 114-118.  doi: 10.1049/trit.2018.0014. [18] W. Zhao, S. Jin and W. Yue, Analysis of the average confirmation time of transactions in a blockchain system, 14th International Conference of Queueing Theory and Network Applications, LNCS, (2019), 379–388. doi: 10.1007/978-3-030-27181-7_23.
The working flow of a mining cycle
Operation of the queueing model with general limited batch service
Change trend of the individual benefit $U_{I}(\lambda)$ of a transaction
Change trend of social benefit $U_{S}(\lambda)$ of transactions
The working flow of the enhanced GOA
Nash equilibrium and socially optimal arrival rates of transactions
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
 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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