Accuracy and precision requirements in probability models
https://doi.org/10.21683/1729-2646-2025-25-4-3-16
Abstract
Numerical Laplace transform and inverse Laplace transform is a challenging task in queueing theory and others probability models. A double transformation approach is used to find stable, accurate, and computationally efficient methods for performing the numerical Laplace and inverse Laplace transform. To validate and improve the inversion solution obtained, direct Laplace transforms are taken of the numerically inverted transforms to compare with the original function. Algorithms provide increasing accuracy as precision level increases. The most promising methods were applied to computational probability models, when there are no closed-form solutions of the Laplace transform inversion. The computational efficiency compared to precision levels is demonstrated for different service models in M/G/1 queuing systems.
About the Author
Z. Krougly
Western University
Канада
Канада
Zinovi Krougly, Department of Applied Mathematics
London, Ontario
References
1. Abate J., Valko P. Multi-precision Laplace transform inversion // International Journal for Numerical Methods in Engineering. 2004. Vol. 60. Pp. 979-993.
2. Atkinson K. An Introduction to Numerical Analysis: 2nd ed. John Wiley & Sons, 1989.
3. Bailey D. High-Precision Software Directory. 2024 URL: https://www.davidhbailey.com/dhbsoftware/ (дата обращения 04.10.2025).
4. Cohen A. Numerical Methods for Laplace Transform Inversion. Springer, 2007.
5. Edwards C., Penney C. Differential Equations and Boundary Value Problems: 5th ed. Computing and Modeling, 2015.
6. Kao E. An Introduction to Stochastic Processes. Duxbury Press, 1997.
7. Kleinrock L. Queueing systems. Volume 1. Moscow: Mashinostriyeniye; 1979.
8. Kreyszig E. Advanced Engineering Mathematics: 10th ed. John Wiley & Sons, 2011. 1280 p.
9. Krougly Z., Davison M., Aiyar S. The role of high precision arithmetic in calculating numerical Laplace and inverse Laplace transforms // Applied Mathematics. 2017. Vol. 8. Pp. 562-589.
10. Krougly Z., Jeffrey D. Implementation and application of extended precision in Matlab / In: N. Mastorakis et al (Eds.). Proc. of the Applied Computing Conference ACC’09, WSEAS Press. 2009. Pp. 103-108.
11. Krougly Z., Stanford D. Iterative algorithms for performance evaluation of closed network models // Performance Evaluation. 2005. Vol. 61. Pp. 41-64.
12. Krougly Z., Jeffrey D., Tsarapkina D. Software implementation of numerical algorithms in arbitrary precision / In: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2013). N. Bjorner et al. (Eds.). IEEE Computer Society. 2014. Pp. 132-138.
13. Kuhlman K. Review of inverse Laplace transform algorithms for Laplace-space numerical approaches // Numerical Algorithms. 2013. Vol. 63. Pp. 339-355.
14. Kuznetsov A. On the convergence of Gaver-Stehfest algorithm // SIAM J. Numer. Anal. 2013. Vol. 51. Pp. 2984-2998.
15. Murli A., Rizzardi M. Algorithm 682 Talbot’s method for the Laplace inversion problem // ACM Transactions on Mathematical Software. 1990. Vol. 16. Pp. 158-168.
16. Nadarajah S., Kotz S. On the Laplace transform of the Pareto distribution // Queueing System. 2006. Vol. 54. Pp. 243-244.
17. Shortle J., Thompson J., Gross D. et al. Fundamentals of Queueing Theory: 5th ed. Wiley, New York, 2018.
18. Stehfest H. Algorithm 368: Numerical inversion of Laplace transforms. Communications of the ACM 1970;13(1):47-49.
Review
For citations:
Krougly Z. Accuracy and precision requirements in probability models. Dependability. 2025;25(4):3-16. (In Russ.) https://doi.org/10.21683/1729-2646-2025-25-4-3-16
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ISSN 1729-2646 (Print)
ISSN 2500-3909 (Online)
ISSN 2500-3909 (Online)
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