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January  2019, 15(1): 365-386. doi: 10.3934/jimo.2018047

Effect of Bitcoin fee on transaction-confirmation process

Shoji Kasahara , and  Jun Kawahara

Graduate School of Information Science, Nara Institute of Science and Technology, Takayama 8916-5, Ikoma, Nara 6300192, Japan

* Corresponding author: Shoji Kasahara

Received  June 2017 Revised  October 2017 Published  January 2019 Early access  April 2018

Fund Project: The first author is supported in part by SCAT Foundation, and Japan Society for the Promotion of Science under Grant-in-Aid for Scientific Research (B) No. 15H04008.

In Bitcoin system, transactions are prioritized according to transaction fees. Transactions without fees are given low priority and likely to wait for confirmation. Because the demand of micro payment in Bitcoin is expected to increase due to low remittance cost, it is important to quantitatively investigate how transactions with small fees of Bitcoin affect the transaction-confirmation time. In this paper, we analyze the transaction-confirmation time by queueing theory. We model the transaction-confirmation process of Bitcoin as a priority queueing system with batch service, deriving the mean transaction-confirmation time. Numerical examples show how the demand of transactions with low fees affects the transaction-confirmation time. We also consider the effect of the maximum block size on the transaction-confirmation time.

Keywords: Bitcoin, blockchain, fee, transaction-confirmation time, priority queue.
Mathematics Subject Classification: Primary: 68M20, 60K25; Secondary: 90B22.
Citation: Shoji Kasahara, Jun Kawahara. Effect of Bitcoin fee on transaction-confirmation process. Journal of Industrial & Management Optimization, 2019, 15 (1) : 365-386. doi: 10.3934/jimo.2018047
References:
[1]

E. Androulaki, G. O. Karame, M. Roeschlin, T. Scherer and S. Capkun, Evaluating user privacy in Bitcoin, The 17th International Conference on Financial Cryptography and Data Security, (2013), 34-51.  doi: 10.1007/978-3-642-39884-1_4.  Google Scholar

[2]

A. M. Antonopoulos, Mastering Bitcoin, O'Reilly, 2014. Google Scholar

[3]

T. Bamert, C. Decker, L. Elsen, R. Wattenhofer and S. Welten, Have a snack, pay with Bitcoins, 2013 IEEE Thirteenth International Conference on Peer-to-Peer Computing, (2013), 1-5.  doi: 10.1109/P2P.2013.6688717.  Google Scholar

[4]

R. Böhme, N. Christin, B. Edelman and T. Moore, Bitcoin: Economics, technology, and governance, Journal of Economic Perspectives, 29 (2015), 213-238.   Google Scholar

[5]

J. Bonneau, A. Miller, J. Clark, A. Narayanan, J. A. Kroll and E. W. Felten, SoK: Research perspectives and challenges for Bitcoin and cryptocurrencies, IEEE Symposium on Security and Privacy, (2015), 104-121.  doi: 10.1109/SP.2015.14.  Google Scholar

[6]

M. L. Chaudhry and J. G. C. Templeton, The queuing system M/$ \mbox{G}^{\text B} $/1 and its ramifications, European Journal of Operational Research, 6 (1981), 56-60.  doi: 10.1016/0377-2217(81)90328-3.  Google Scholar

[7]

M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, John Wiley & Sons, 1983.  Google Scholar

[8]

C. Decker and R. Wattenhofer, Information propagation in the Bitcoin network, 13th IEEE International Conference on Peer-to-Peer Computing, (2013), 1-10.  doi: 10.1109/P2P.2013.6688704.  Google Scholar

[9]

J. Göbel, H. P. Keeler, A. E. Krzesinski and P. G. Taylor, Bitcoin blockchain dynamics: The selfish-mine strategy in the presence of propagation delay, Performance Evaluation, 104 (2016), 23-41.   Google Scholar

[10]

G. O. Karame, E. Androulaki and S. Capkun, Double-spending fast payments in Bitcoin, The 2012 ACM Conference on Computer and Communications Security, (2012), 906-917.  doi: 10.1145/2382196.2382292.  Google Scholar

[11]

A. Kiayias and G. Panagiotakos, Speed-security tradeoffs in blockchain protocols, IACR: Cryptology ePrint Archive, 2015. Google Scholar

[12]

S. Kotz and S. Nadarajah, Extreme Value Distributions Theory and Applications, Imperial College Press, 2000.  Google Scholar

[13]

M. Möser and R. Böhome, Trends, tips, tolls: A longitudinal study of Bitcoin transaction fees, Financial Cryptography and Data Security, Lecture Notes in Computer Science, Springer, 8976 (2015), 19-33.  Google Scholar

[14]

S. Nakamoto, Bitcoin: A peer-to-peer electronic cash system, (2008). Available from https://bitcoin.org/bitcoin.pdf. Google Scholar

[15]

R. Peter, A transaction fee market exists without a block size limit, (2015). Available from https://scalingbitcoin.org/papers/feemarket.pdf Google Scholar

[16]

Y. Sompolinsky and A. Zohar, Accelerating Bitcoin's transaction processing. Fast money grows on trees, not chains, IACR: Cryptology ePrint Archive, 2013, Available from https://eprint.iacr.org/2013/881. Google Scholar

[17]

Y. Sompolinsky and A. Zohar, Secure high-rate transaction processing in Bitcoin, 19th International Conference on Financial Cryptography and Data Security, 8975 (2015), 507-527.   Google Scholar

[18]

H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, North-Holland Publishing Co., Amsterdam, 1993.  Google Scholar

[19]

F. Tschorsch and B. Scheuermann, Bitcoin and beyond: A technical survey on decentralized digital currencies, IEEE Communications Surveys & Tutorials, 18 (2016), 2084-2123.  doi: 10.1109/COMST.2016.2535718.  Google Scholar

[20]

R. W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice Hall, 1989.  Google Scholar

[21]

https://bitnodes.21.co/. Google Scholar

[22]

https://blockchain.info/. Google Scholar

[23]

https://cdecker.github.io/btcresearch/. Google Scholar

[24]

https://en.bitcoin.it/wiki/Confirmation. Google Scholar

[25]

https://en.bitcoin.it/wiki/Scalability. Google Scholar

[26]

https://en.bitcoin.it/wiki/Transaction_fees. Google Scholar

show all references

References:
[1]

E. Androulaki, G. O. Karame, M. Roeschlin, T. Scherer and S. Capkun, Evaluating user privacy in Bitcoin, The 17th International Conference on Financial Cryptography and Data Security, (2013), 34-51.  doi: 10.1007/978-3-642-39884-1_4.  Google Scholar

[2]

A. M. Antonopoulos, Mastering Bitcoin, O'Reilly, 2014. Google Scholar

[3]

T. Bamert, C. Decker, L. Elsen, R. Wattenhofer and S. Welten, Have a snack, pay with Bitcoins, 2013 IEEE Thirteenth International Conference on Peer-to-Peer Computing, (2013), 1-5.  doi: 10.1109/P2P.2013.6688717.  Google Scholar

[4]

R. Böhme, N. Christin, B. Edelman and T. Moore, Bitcoin: Economics, technology, and governance, Journal of Economic Perspectives, 29 (2015), 213-238.   Google Scholar

[5]

J. Bonneau, A. Miller, J. Clark, A. Narayanan, J. A. Kroll and E. W. Felten, SoK: Research perspectives and challenges for Bitcoin and cryptocurrencies, IEEE Symposium on Security and Privacy, (2015), 104-121.  doi: 10.1109/SP.2015.14.  Google Scholar

[6]

M. L. Chaudhry and J. G. C. Templeton, The queuing system M/$ \mbox{G}^{\text B} $/1 and its ramifications, European Journal of Operational Research, 6 (1981), 56-60.  doi: 10.1016/0377-2217(81)90328-3.  Google Scholar

[7]

M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, John Wiley & Sons, 1983.  Google Scholar

[8]

C. Decker and R. Wattenhofer, Information propagation in the Bitcoin network, 13th IEEE International Conference on Peer-to-Peer Computing, (2013), 1-10.  doi: 10.1109/P2P.2013.6688704.  Google Scholar

[9]

J. Göbel, H. P. Keeler, A. E. Krzesinski and P. G. Taylor, Bitcoin blockchain dynamics: The selfish-mine strategy in the presence of propagation delay, Performance Evaluation, 104 (2016), 23-41.   Google Scholar

[10]

G. O. Karame, E. Androulaki and S. Capkun, Double-spending fast payments in Bitcoin, The 2012 ACM Conference on Computer and Communications Security, (2012), 906-917.  doi: 10.1145/2382196.2382292.  Google Scholar

[11]

A. Kiayias and G. Panagiotakos, Speed-security tradeoffs in blockchain protocols, IACR: Cryptology ePrint Archive, 2015. Google Scholar

[12]

S. Kotz and S. Nadarajah, Extreme Value Distributions Theory and Applications, Imperial College Press, 2000.  Google Scholar

[13]

M. Möser and R. Böhome, Trends, tips, tolls: A longitudinal study of Bitcoin transaction fees, Financial Cryptography and Data Security, Lecture Notes in Computer Science, Springer, 8976 (2015), 19-33.  Google Scholar

[14]

S. Nakamoto, Bitcoin: A peer-to-peer electronic cash system, (2008). Available from https://bitcoin.org/bitcoin.pdf. Google Scholar

[15]

R. Peter, A transaction fee market exists without a block size limit, (2015). Available from https://scalingbitcoin.org/papers/feemarket.pdf Google Scholar

[16]

Y. Sompolinsky and A. Zohar, Accelerating Bitcoin's transaction processing. Fast money grows on trees, not chains, IACR: Cryptology ePrint Archive, 2013, Available from https://eprint.iacr.org/2013/881. Google Scholar

[17]

Y. Sompolinsky and A. Zohar, Secure high-rate transaction processing in Bitcoin, 19th International Conference on Financial Cryptography and Data Security, 8975 (2015), 507-527.   Google Scholar

[18]

H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, North-Holland Publishing Co., Amsterdam, 1993.  Google Scholar

[19]

F. Tschorsch and B. Scheuermann, Bitcoin and beyond: A technical survey on decentralized digital currencies, IEEE Communications Surveys & Tutorials, 18 (2016), 2084-2123.  doi: 10.1109/COMST.2016.2535718.  Google Scholar

[20]

R. W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice Hall, 1989.  Google Scholar

[21]

https://bitnodes.21.co/. Google Scholar

[22]

https://blockchain.info/. Google Scholar

[23]

https://cdecker.github.io/btcresearch/. Google Scholar

[24]

https://en.bitcoin.it/wiki/Confirmation. Google Scholar

[25]

https://en.bitcoin.it/wiki/Scalability. Google Scholar

[26]

https://en.bitcoin.it/wiki/Transaction_fees. Google Scholar

Figure 1.  Trend of fee-amount distribution over time
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Figure 2.  Trend of transaction-arrival rates of two priority classes
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Figure 3.  Relative frequency and exponential probability density function of block-generation time
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Figure 4.  Comparison of analysis and simulation for the transaction-confirmation time: Two-priority case
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Figure 5.  Mean transaction-confirmation time: classless case
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Figure 6.  Mean transaction-confirmation time: two-priority case. ($\lambda_H = 0.90466$)
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Figure 7.  Mean transaction-confirmation time: high priority case. The ratio of $\lambda_H$ to $\lambda_L$ is fixed, and the overall arrival rate $\lambda$ changes
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Figure 8.  Mean transaction-confirmation time: low priority case. The ratio of $\lambda_H$ to $\lambda_L$ is fixed, and the overall arrival rate $\lambda$ changes
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Table 1.  Block-generation time
Mean [s]544.09
Variance $2.9277 \times 10^{5}$
Maximum [s]6,524
Minimum [s]0
Median [s]377
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Mean [s]544.09
Variance $2.9277 \times 10^{5}$
Maximum [s]6,524
Minimum [s]0
Median [s]377
Table 2.  Number of transactions in a block
Mean [transactions]529.27
Variance $2.5152 \times 10^5$
Maximum [transactions]12,239
Minimum [transactions]0
Median [transactions]386
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Mean [transactions]529.27
Variance $2.5152 \times 10^5$
Maximum [transactions]12,239
Minimum [transactions]0
Median [transactions]386
Table 3.  Transaction size in byte
Mean571.34
Variance $3.7445\times 10^6$
Maximum999657
Minimum62
Median259
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Mean571.34
Variance $3.7445\times 10^6$
Maximum999657
Minimum62
Median259
Table 4.  Cumulative frequency of fee amount for transactions
BTCFrequency
01378501
0.000013050709
0.000142881857
0.00160723356
0.0161219997
0.161236481
161236972
1061237045
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BTCFrequency
01378501
0.000013050709
0.000142881857
0.00160723356
0.0161219997
0.161236481
161236972
1061237045
Table 5.  Transaction-type statistics
StatisticClasslessHL
Number of transactions61,353,01457,058,9474,294,067
Mean TCT [s]1075.0874.133744.1
Variance of TCT $1.8989 \times 10^8$ $8.4505 \times 10^7$ $1.5826 \times 10^9$
Maximum of TCT $3.1045\times 10^7$ $3.1045\times 10^7$ $2.6244\times 10^7$
Minimum of TCT000
Median of TCT510502640
Mean arrival rate0.972750.904660.068082
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StatisticClasslessHL
Number of transactions61,353,01457,058,9474,294,067
Mean TCT [s]1075.0874.133744.1
Variance of TCT $1.8989 \times 10^8$ $8.4505 \times 10^7$ $1.5826 \times 10^9$
Maximum of TCT $3.1045\times 10^7$ $3.1045\times 10^7$ $2.6244\times 10^7$
Minimum of TCT000
Median of TCT510502640
Mean arrival rate0.972750.904660.068082
Table 6.  Comparison of analysis and measurement for the transaction-confirmation time
Transaction TypeArrival RateMeasurementAnalysis
Classless0.972751,075.0568.10
H0.90466874.13562.16
L0.0680823,744.1647.05
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Transaction TypeArrival RateMeasurementAnalysis
Classless0.972751,075.0568.10
H0.90466874.13562.16
L0.0680823,744.1647.05
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