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

Wenjuan Zhao; Shunfu Jin; Wuyi Yue

doi:10.3934/jimo.2020049

Article Contents

2021, Volume 17, Issue 4: 1845-1861. Doi: 10.3934/jimo.2020049

This issue Previous Article The viability of switched nonlinear systems with piecewise smooth Lyapunov functions Next Article Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints

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
^* Corresponding author: Shunfu Jin

Received: April 30, 2019

Revised: September 30, 2019

Early access: March 2020

Published: July 2021

Abstract Full Text(HTML) Figure(6) / Table(1) Related Papers Cited by

Abstract

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:

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

Citation:

FullText(HTML)

Figure 1. The working flow of a mining cycle

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

Figure 5. The working flow of the enhanced GOA

Download: Full-size image PowerPoint slide

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

Download: Full-size image 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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.