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Comparison of analysis and simulation for basic queueing model
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doi: 10.3934/jimo.2018193
Priority queueing analysis of transaction-confirmation time for Bitcoin
Graduate School of Information Science, Nara Institute of Science and Technology, Takayama 8916-5, Ikoma, Nara 6300192, Japan |
In Bitcoin system, a transaction is given a priority value according to its attributes such as the remittance amount and fee, and transactions with high priorities are likely to be confirmed faster than those with low priorities. In this paper, we analyze the transaction-confirmation time for Bitcoin system. We model the transaction-confirmation process as a queueing system with batch service, M/$ \mbox{G}^B $/1. We consider the joint distribution of numbers of transactions in system and the elapsed service time, deriving the mean transaction-confirmation time. Using the result, we derive the recursive formulae of mean transaction-confirmation times of an M/$ \mbox{G}^B $/1 queue with priority service discipline. In numerical examples, we show the effect of the maximum block size on the mean transaction-confirmation time, investigating the accuracy region of our queueing model. We also discuss how the increase in micropayments, which are likely to be given low priorities, affects the transaction-confirmation time.
Mathematics Subject Classification:
Primary: 68M20, 90B22; Secondary: 60K25.
Citation:
Yoshiaki Kawase, Shoji Kasahara.
Priority queueing analysis of transaction-confirmation time for Bitcoin. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018193
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References:
| [1] |
A. M. Antonopoulos, Mastering Bitcoin, O'Reilly, 2014. |
| [2] |
M. L. Chaudhry and J. G. C. Templeton,
The queuing system M/GB/1 and its ramifications, European Journal of Operational Research, 6 (1981), 56-60.
doi: 10.1016/0377-2217(81)90328-3. |
| [3] |
M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, John Wiley & Sons, 1983. |
| [4] |
http://www.meti.go.jp/committee/kenkyukai/sansei/fintech\_kadai/pdf/003\_02\_00.pdf |
| [5] | |
| [6] | |
| [7] |
http://www.coindesk.com/segregated-witness-bitcoin-block-size-debate/ |
| [8] |
S. Kasahara and J. Kawahara, Effect of Bitcoin fee on transaction-confirmation process, arXiv: 1604.00103[cs.CR]. |
| [9] |
Y. Kawase and S. Kasahara, Transaction-Confirmation Time for Bitcoin: A Queueing Analytical Approach to Blockchain Mechanism, The 12th International Conference on Queueing Theory and Network Applications (QTNA2017), Qinhuangdao, China, August 21-23, 2017. |
| [10] |
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.
doi: 10.1007/978-3-662-48051-9_2. |
| [11] |
S. Nakamoto, Bitcoin: A Peer-to-Peer Electronic Cash System, 2008. Available from: https://bitcoin.org/bitcoin.pdf. |
| [12] |
J. Poon and T. Dryja, The bitcoin lightning network: Scalable off-chain instant payments, https://lightning.network/lightning-network-paper.pdf, 2016. |
| [13] |
H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Vol. 2. Finite systems. North-Holland Publishing Co., Amsterdam, 1993. |
| [14] |
https://github.com/bitcoin/bips/blob/master/bip-0141.mediawiki |
| [15] |
F. Tschorsch and B. Scheuermann, Bitcoin and beyond: A technical survey on decentralized digital currencies, Tutorials, 18 (2016), 2084-2123. |
| [16] | |
| [17] |
https://www.coindesk.com/segwits-slow-rollout-bitcoins-capacity-hasnt-seen/ |
show all references
References:
| [1] |
A. M. Antonopoulos, Mastering Bitcoin, O'Reilly, 2014. |
| [2] |
M. L. Chaudhry and J. G. C. Templeton,
The queuing system M/GB/1 and its ramifications, European Journal of Operational Research, 6 (1981), 56-60.
doi: 10.1016/0377-2217(81)90328-3. |
| [3] |
M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, John Wiley & Sons, 1983. |
| [4] |
http://www.meti.go.jp/committee/kenkyukai/sansei/fintech\_kadai/pdf/003\_02\_00.pdf |
| [5] | |
| [6] | |
| [7] |
http://www.coindesk.com/segregated-witness-bitcoin-block-size-debate/ |
| [8] |
S. Kasahara and J. Kawahara, Effect of Bitcoin fee on transaction-confirmation process, arXiv: 1604.00103[cs.CR]. |
| [9] |
Y. Kawase and S. Kasahara, Transaction-Confirmation Time for Bitcoin: A Queueing Analytical Approach to Blockchain Mechanism, The 12th International Conference on Queueing Theory and Network Applications (QTNA2017), Qinhuangdao, China, August 21-23, 2017. |
| [10] |
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.
doi: 10.1007/978-3-662-48051-9_2. |
| [11] |
S. Nakamoto, Bitcoin: A Peer-to-Peer Electronic Cash System, 2008. Available from: https://bitcoin.org/bitcoin.pdf. |
| [12] |
J. Poon and T. Dryja, The bitcoin lightning network: Scalable off-chain instant payments, https://lightning.network/lightning-network-paper.pdf, 2016. |
| [13] |
H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Vol. 2. Finite systems. North-Holland Publishing Co., Amsterdam, 1993. |
| [14] |
https://github.com/bitcoin/bips/blob/master/bip-0141.mediawiki |
| [15] |
F. Tschorsch and B. Scheuermann, Bitcoin and beyond: A technical survey on decentralized digital currencies, Tutorials, 18 (2016), 2084-2123. |
| [16] | |
| [17] |
https://www.coindesk.com/segwits-slow-rollout-bitcoins-capacity-hasnt-seen/ |
Figure 2.
Comparison of analysis and simulation for priority-queueing model
Figure 3.
Transaction-confirmation time vs. block size, 1st period (October 2013 to September 2014). $ \lambda = 0.7336929 $ , $ \mu = 0.0019748858 $ , and the coefficient of variation of transaction inter-arrival time: $ 3.72401599 $
Figure 4.
Transaction-confirmation time vs. block size, 2nd period (October 2014 to September 2015). $ \lambda = 1.2081311 $ , $ \mu = 0.0017009449 $ , and the coefficient of variation of transaction inter-arrival time: $ 15.3250509 $
Figure 5.
The mean transaction-arrival rate
Figure 6.
Transaction-confirmation time vs. block size, October 2013 to September 2014. $ \lambda_H = 0.7203544 $ , $ \mu = 0.0019748858 $
Figure 7.
Transaction-confirmation time vs. block size, October 2014 to September 2015. $ \lambda_H = 1.1494675 $ , $ \mu = 0.0017009449 $
Figure 8.
Transaction-confirmation time vs. block size, October 2013 to September 2014. $ \lambda_L = 0.0133385 $ , $ \mu = 0.0019748858 $
Figure 9.
Transaction-confirmation time vs. block size, October 2014 to September 2015. $ \lambda_L = 0.0586636 $ , $ \mu = 0.0017009449 $
Figure 10.
The effects of the block size on the transaction-confirmation time
Figure 11.
The effects of the micropayments on the transaction-confirmation time
Figure 12.
The effect of the block size on the transaction-confirmation time of high-priority transactions
Figure 13.
The effects of the block size on the transaction-confirmation time of low-priority transactions
Table 1.
Mean transaction confirmation time for each priority
| Period | Classless [s] | Low priority [s] | High priority [s] | |
| 1st | (2013/10/1-2014/9/30) | 1060.67797 | 1947.47115 | 1044.25043 |
| 2nd | (2014/10/1-2015/9/30) | 1171.98866 | 4887.56496 | 1070.32818 |
| overall | (2013/10/1-2015/9/30) | 1124.13286 | 3888.06977 | 1059.06133 |
| Period | Classless [s] | Low priority [s] | High priority [s] | |
| 1st | (2013/10/1-2014/9/30) | 1060.67797 | 1947.47115 | 1044.25043 |
| 2nd | (2014/10/1-2015/9/30) | 1171.98866 | 4887.56496 | 1070.32818 |
| overall | (2013/10/1-2015/9/30) | 1124.13286 | 3888.06977 | 1059.06133 |
Table 2.
Comparison of analysis and measurement
| Transaction Type | Arrival Rate | Measurement [s] | Analysis[s] |
| Classless | 0.9709120529 | 1124.13286 | 1112.025339 |
| High Priority | 0.9349109906 | 1059.06133 | 1108.773595 |
| Low Priority | 0.0360010622 | 3888.06977 | 1196.469817 |
| Transaction Type | Arrival Rate | Measurement [s] | Analysis[s] |
| Classless | 0.9709120529 | 1124.13286 | 1112.025339 |
| High Priority | 0.9349109906 | 1059.06133 | 1108.773595 |
| Low Priority | 0.0360010622 | 3888.06977 | 1196.469817 |
Table 3.
Coefficients of variation for transaction interarrival time
| Period | 1st 2013/10-2014/09 |
2nd 2014/10-2015/09 |
Overall 2013/10-2015/09 |
| Classless | 3.72401 | 15.32505 | 10.17893 |
| Period | 1st 2013/10-2014/09 |
2nd 2014/10-2015/09 |
Overall 2013/10-2015/09 |
| Classless | 3.72401 | 15.32505 | 10.17893 |
Table 4.
Coefficients of variation for transaction interarrival time, two-priority classes
| Period | 1st 2013/10-2014/09 |
2nd 2014/10-2015/09 |
Overall 2013/10-2015/09 |
| Low Priority | 1.66569 | 3.90654 | 3.01387 |
| High Priority | 3.70045 | 14.94895 | 9.99103 |
| Period | 1st 2013/10-2014/09 |
2nd 2014/10-2015/09 |
Overall 2013/10-2015/09 |
| Low Priority | 1.66569 | 3.90654 | 3.01387 |
| High Priority | 3.70045 | 14.94895 | 9.99103 |
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