# American Institute of Mathematical Sciences

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

## Effect of Bitcoin fee on transaction-confirmation process

 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  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.

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:
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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] [22] [23] [24] [25] [26]
Trend of fee-amount distribution over time
Trend of transaction-arrival rates of two priority classes
Relative frequency and exponential probability density function of block-generation time
Comparison of analysis and simulation for the transaction-confirmation time: Two-priority case
Mean transaction-confirmation time: classless case
Mean transaction-confirmation time: two-priority case. ($\lambda_H = 0.90466$)
Mean transaction-confirmation time: high priority case. The ratio of $\lambda_H$ to $\lambda_L$ is fixed, and the overall arrival rate $\lambda$ changes
Mean transaction-confirmation time: low priority case. The ratio of $\lambda_H$ to $\lambda_L$ is fixed, and the overall arrival rate $\lambda$ changes
Block-generation time
 Mean [s] 544.09 Variance $2.9277 \times 10^{5}$ Maximum [s] 6,524 Minimum [s] 0 Median [s] 377
 Mean [s] 544.09 Variance $2.9277 \times 10^{5}$ Maximum [s] 6,524 Minimum [s] 0 Median [s] 377
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
 Mean [transactions] 529.27 Variance $2.5152 \times 10^5$ Maximum [transactions] 12,239 Minimum [transactions] 0 Median [transactions] 386
Transaction size in byte
 Mean 571.34 Variance $3.7445\times 10^6$ Maximum 999657 Minimum 62 Median 259
 Mean 571.34 Variance $3.7445\times 10^6$ Maximum 999657 Minimum 62 Median 259
Cumulative frequency of fee amount for transactions
 BTC Frequency 0 1378501 0.00001 3050709 0.0001 42881857 0.001 60723356 0.01 61219997 0.1 61236481 1 61236972 10 61237045
 BTC Frequency 0 1378501 0.00001 3050709 0.0001 42881857 0.001 60723356 0.01 61219997 0.1 61236481 1 61236972 10 61237045
Transaction-type statistics
 Statistic Classless H L Number of transactions 61,353,014 57,058,947 4,294,067 Mean TCT [s] 1075.0 874.13 3744.1 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 TCT 0 0 0 Median of TCT 510 502 640 Mean arrival rate 0.97275 0.90466 0.068082
 Statistic Classless H L Number of transactions 61,353,014 57,058,947 4,294,067 Mean TCT [s] 1075.0 874.13 3744.1 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 TCT 0 0 0 Median of TCT 510 502 640 Mean arrival rate 0.97275 0.90466 0.068082
Comparison of analysis and measurement for the transaction-confirmation time
 Transaction Type Arrival Rate Measurement Analysis Classless 0.97275 1,075.0 568.10 H 0.90466 874.13 562.16 L 0.068082 3,744.1 647.05
 Transaction Type Arrival Rate Measurement Analysis Classless 0.97275 1,075.0 568.10 H 0.90466 874.13 562.16 L 0.068082 3,744.1 647.05
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