# American Institute of Mathematical Sciences

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

Received  October 2017 Revised  April 2018 Published  December 2018

Fund Project: The second author is supported by Grant-in-Aid for Scientific Research (B) No. 15H04008 and SCAT Foundation

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.

Citation: Yoshiaki Kawase, Shoji Kasahara. Priority queueing analysis of transaction-confirmation time for Bitcoin. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018193
##### References:
 [1] A. M. Antonopoulos, Mastering Bitcoin, O'Reilly, 2014. Google Scholar [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.  Google Scholar [3] M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, John Wiley & Sons, 1983.  Google Scholar [4] [5] [6] [7] [8] S. Kasahara and J. Kawahara, Effect of Bitcoin fee on transaction-confirmation process, arXiv: 1604.00103[cs.CR]. Google Scholar [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. Google Scholar [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.  Google Scholar [11] S. Nakamoto, Bitcoin: A Peer-to-Peer Electronic Cash System, 2008. Available from: https://bitcoin.org/bitcoin.pdf. Google Scholar [12] J. Poon and T. Dryja, The bitcoin lightning network: Scalable off-chain instant payments, https://lightning.network/lightning-network-paper.pdf, 2016. Google Scholar [13] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Vol. 2. Finite systems. North-Holland Publishing Co., Amsterdam, 1993.  Google Scholar [14] [15] F. Tschorsch and B. Scheuermann, Bitcoin and beyond: A technical survey on decentralized digital currencies, Tutorials, 18 (2016), 2084-2123.   Google Scholar [16] [17]

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##### References:
 [1] A. M. Antonopoulos, Mastering Bitcoin, O'Reilly, 2014. Google Scholar [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.  Google Scholar [3] M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, John Wiley & Sons, 1983.  Google Scholar [4] [5] [6] [7] [8] S. Kasahara and J. Kawahara, Effect of Bitcoin fee on transaction-confirmation process, arXiv: 1604.00103[cs.CR]. Google Scholar [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. Google Scholar [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.  Google Scholar [11] S. Nakamoto, Bitcoin: A Peer-to-Peer Electronic Cash System, 2008. Available from: https://bitcoin.org/bitcoin.pdf. Google Scholar [12] J. Poon and T. Dryja, The bitcoin lightning network: Scalable off-chain instant payments, https://lightning.network/lightning-network-paper.pdf, 2016. Google Scholar [13] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Vol. 2. Finite systems. North-Holland Publishing Co., Amsterdam, 1993.  Google Scholar [14] [15] F. Tschorsch and B. Scheuermann, Bitcoin and beyond: A technical survey on decentralized digital currencies, Tutorials, 18 (2016), 2084-2123.   Google Scholar [16] [17]
Comparison of analysis and simulation for basic queueing model
Comparison of analysis and simulation for priority-queueing model
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: $3.72401599$
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: $15.3250509$
The mean transaction-arrival rate
Transaction-confirmation time vs. block size, October 2013 to September 2014. $\lambda_H = 0.7203544$, $\mu = 0.0019748858$
Transaction-confirmation time vs. block size, October 2014 to September 2015. $\lambda_H = 1.1494675$, $\mu = 0.0017009449$
Transaction-confirmation time vs. block size, October 2013 to September 2014. $\lambda_L = 0.0133385$, $\mu = 0.0019748858$
Transaction-confirmation time vs. block size, October 2014 to September 2015. $\lambda_L = 0.0586636$, $\mu = 0.0017009449$
The effects of the block size on the transaction-confirmation time
The effects of the micropayments on the transaction-confirmation time
The effect of the block size on the transaction-confirmation time of high-priority transactions
The effects of the block size on the transaction-confirmation time of low-priority transactions
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
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
Coefficients of variation for transaction interarrival time
 Period 1st2013/10-2014/09 2nd2014/10-2015/09 Overall2013/10-2015/09 Classless 3.72401 15.32505 10.17893
 Period 1st2013/10-2014/09 2nd2014/10-2015/09 Overall2013/10-2015/09 Classless 3.72401 15.32505 10.17893
Coefficients of variation for transaction interarrival time, two-priority classes
 Period 1st2013/10-2014/09 2nd2014/10-2015/09 Overall2013/10-2015/09 Low Priority 1.66569 3.90654 3.01387 High Priority 3.70045 14.94895 9.99103
 Period 1st2013/10-2014/09 2nd2014/10-2015/09 Overall2013/10-2015/09 Low Priority 1.66569 3.90654 3.01387 High Priority 3.70045 14.94895 9.99103
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