American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2020049

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

Wenjuan Zhao 1, , Shunfu Jin 1,, and  Wuyi Yue 2,

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  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.

Keywords: Blockchain, limited batch service, Markov chain, Nash equilibrium, social optimization, intelligent optimization algorithm, pricing policy.
Mathematics Subject Classification: Primary: 60K15, 60K25; Secondary: 68M14.
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 & Management Optimization, 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.  Google Scholar

[2]

A. Antonopoulos and O. Media, Mastering Bitcoin: Unlocking Digital Crypto-Currencies, O'Reilly Media, 2014. Google Scholar

[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.  Google Scholar

[4]

S. Kasahara and J. Kawahara, Effect on transaction-confirmation process, J. Ind. Manag. Optim., 15 (2019), 365-386.   Google Scholar

[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.  Google Scholar

[6]

Q. Li, J. Ma and Y. Chang, Blockchain queueing theory, (2018). Available from: https://arXiv.org/abs/1808.01795. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

S. Ross, Stochastic Processes, Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1996.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

M. Vlasiou, A non-increasing Lindley-type equation, Queueing Systems, 56 (2007), 41-52.  doi: 10.1007/s11134-007-9029-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. Antonopoulos and O. Media, Mastering Bitcoin: Unlocking Digital Crypto-Currencies, O'Reilly Media, 2014. Google Scholar

[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.  Google Scholar

[4]

S. Kasahara and J. Kawahara, Effect on transaction-confirmation process, J. Ind. Manag. Optim., 15 (2019), 365-386.   Google Scholar

[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.  Google Scholar

[6]

Q. Li, J. Ma and Y. Chang, Blockchain queueing theory, (2018). Available from: https://arXiv.org/abs/1808.01795. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

S. Ross, Stochastic Processes, Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1996.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

M. Vlasiou, A non-increasing Lindley-type equation, Queueing Systems, 56 (2007), 41-52.  doi: 10.1007/s11134-007-9029-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The working flow of a mining cycle
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 5.  The working flow of the enhanced GOA
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
[1]

Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122
[2]

Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005
[3]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017
[4]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172
[5]

Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167
[6]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050
[7]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115
[8]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098
[9]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176
[10]

Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381
[11]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031
[12]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[13]

Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109
[14]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
[15]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100
[16]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117
[17]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139
[18]

C. J. Price. A modified Nelder-Mead barrier method for constrained optimization. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020058
[19]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391
[20]

Hanyu Gu, Hue Chi Lam, Yakov Zinder. Planning rolling stock maintenance: Optimization of train arrival dates at a maintenance center. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020177

2019 Impact Factor: 1.366

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences