Properties of nonlinear noise in long, dispersion-uncompensated fiber links |
Optics Express, Vol. 21, Issue 22, pp. 25685-25699 (2013)
http://dx.doi.org/10.1364/OE.21.025685
Acrobat PDF (2111 KB)
Abstract
We study the properties of nonlinear interference noise (NLIN) in fiber-optic communications systems with large accumulated dispersion. Our focus is on settling the discrepancy between the results of the Gaussian noise (GN) model (according to which NLIN is additive Gaussian) and a recently published time-domain analysis, which attributes drastically different properties to the NLIN. Upon reviewing the two approaches we identify several unjustified assumptions that are key in the derivation of the GN model, and that are responsible for the discrepancy. We derive the true NLIN power and verify that the NLIN is not additive Gaussian, but rather it depends strongly on the data transmitted in the channel of interest. In addition we validate the time-domain model numerically and demonstrate the strong dependence of the NLIN on the interfering channels’ modulation format.
© 2013 Optical Society of America
1. Introduction
1. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000). [CrossRef]
3. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008). [CrossRef]
4. R. I. Killey, P. M. Watts, V. Mikhailov, M. Glick, and P. Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. , 17, 714–716(2005). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
12. P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” IEEE J. of Lightwave Technol. 31, 1273–1282 (2013). [CrossRef]
14. G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012) [CrossRef]
16. S. J. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photon. Technol. Lett. 25, 961–964, (2013) [CrossRef]
17. A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef]
17. A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
18. X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013). [CrossRef]
17. A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef]
17. A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
12. P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” IEEE J. of Lightwave Technol. 31, 1273–1282 (2013). [CrossRef]
2. Time-domain analysis
21. S. Kumar and D. Yang, “Second-order theory for self-phase modulation andcross-phase modulation in optical fibers,” J. of Lightwave Technol. 23, 2073–2080, (2005) [CrossRef]
17. A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef]
26. A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13(1994). [CrossRef]
24. X. Zhou, “Hardware efficient carrier recovery algorithms for single-carrier QAM systems,” in Advanced Photonics Congress, OSA Technical Digest (online)(Optical Society of America, 2012), paper SpTu3A.1. [CrossRef]
3. Frequency domain analysis
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
4. Numerical validation
4.1. Modulation format dependence
17. A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
12. P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” IEEE J. of Lightwave Technol. 31, 1273–1282 (2013). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
4.2. The variance of phase-noise and assessment of the residual NLIN
17. A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef]
4.3. The difference with respect to the NLIN power predicted by the GN model
5. Discussion
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
6. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
7. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
24. X. Zhou, “Hardware efficient carrier recovery algorithms for single-carrier QAM systems,” in Advanced Photonics Congress, OSA Technical Digest (online)(Optical Society of America, 2012), paper SpTu3A.1. [CrossRef]
Acknowledgment
References and links
1. | A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000). [CrossRef] |
2. | F. Forghieri, R.W. Tkack, and A. R. Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” Ch. 8 in Optical Fiber Telecommunications IIIA, P. Kaminow and T. L. Koch, eds. (Academic Press, 1997). |
3. | E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008). [CrossRef] |
4. | R. I. Killey, P. M. Watts, V. Mikhailov, M. Glick, and P. Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett. , 17, 714–716(2005). [CrossRef] |
5. | A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified fiber communication systems taking into account fiber nonlinearities,” in 19th European Conference on Optical Communication (ECOC), ECOC Technical Digest, (1993), Paper MoC2.4. |
6. | P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. , 23, 742–744, (2011). [CrossRef] |
7. | A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef] |
8. | P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics]. |
9. | P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. of Light-wave Technol. 30, 3857–3879, (2012). [CrossRef] |
10. | A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. Tapia Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07. |
11. | E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2. |
12. | P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” IEEE J. of Lightwave Technol. 31, 1273–1282 (2013). [CrossRef] |
13. | A. Bononi and P. Serena, “An alternative derivation of Johannissons regular perturbation model,” July19, 2012[Online]. Available: arXiv:1207.4729v1 [physics.optics]. |
14. | G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, and P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012) [CrossRef] |
15. | A. Bononi, N. Rossi, and P. Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” Proceeding of the European Conf. on Opt. Comm. (ECOC), Paper We.2.C.4, Amsterdam (2012) |
16. | S. J. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photon. Technol. Lett. 25, 961–964, (2013) [CrossRef] |
17. | A. Mecozzi and R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef] |
18. | X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013). [CrossRef] |
19. | R. Dar, M. Shtaif, and M. Feder, “New bounds on the capacity of fiber-optics communications,” arXiv:1305.1762v1 [physics.optics]. |
20. | R. Dar, M. Shtaif, and M. Feder, “Improved bounds on the nonlinear fiber-channel capacity,” in Proc. ECOC, 2013, paper P.4.16. |
21. | S. Kumar and D. Yang, “Second-order theory for self-phase modulation andcross-phase modulation in optical fibers,” J. of Lightwave Technol. 23, 2073–2080, (2005) [CrossRef] |
22. | We have assumed a fixed position independent dispersion parameter β″. The generalization to position dependent dispersion is straightforward, but we avoid it as it considerably complicates the notation. |
23. | Although terms involving X_{0,k,m}with m≠ k also contribute to phase noise, they are much smaller than the terms proportional to X_{0,m,m}(particularly in the limit of large dispersion) and we omit them from the analysis presented in this paper. |
24. | X. Zhou, “Hardware efficient carrier recovery algorithms for single-carrier QAM systems,” in Advanced Photonics Congress, OSA Technical Digest (online)(Optical Society of America, 2012), paper SpTu3A.1. [CrossRef] |
25. | A. Tolmachev, I. Tselniker, M. Meltsin, I. Sigron, and M. Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8 |
26. | A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13(1994). [CrossRef] |
27. | R.-J. Essiambre, G. Kramer, P.J. Winzer, G.J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701, (2010). [CrossRef] |
28. | S. Asmussen and P.W. Glynn, Stochastic Simulation, Algorithms and Analysis (Springer, 2007). |
29. | The numerical curves in Figs. 3 and 4 reproduce the plots reported in [20], but with larger statsitics. |
OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems
ToC Category:
Fiber Optics and Optical Communications
History
Original Manuscript: July 29, 2013
Revised Manuscript: September 7, 2013
Manuscript Accepted: September 16, 2013
Published: October 21, 2013
Citation
Ronen Dar, Meir Feder, Antonio Mecozzi, and Mark Shtaif, "Properties of nonlinear noise in long, dispersion-uncompensated fiber
links," Opt. Express 21, 25685-25699 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-25685
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References
- A. Mecozzi, C. B. Clausen, M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394(2000). [CrossRef]
- F. Forghieri, R.W. Tkack, A. R. Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” Ch. 8 in Optical Fiber Telecommunications IIIA, P. Kaminow, T. L. Koch, eds. (Academic Press, 1997).
- E. Ip, J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008). [CrossRef]
- R. I. Killey, P. M. Watts, V. Mikhailov, M. Glick, P. Bayvel, “Electronic dispersion compensation by signal predistortion using digital processing and a dual-drive Mach-Zehnder modulator,” IEEE Photon. Technol. Lett., 17, 714–716(2005). [CrossRef]
- A. Splett, C. Kurtzke, K. Petermann, “Ultimate transmission capacity of amplified fiber communication systems taking into account fiber nonlinearities,” in 19th European Conference on Optical Communication (ECOC), ECOC Technical Digest, (1993), Paper MoC2.4.
- P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett., 23, 742–744, (2011). [CrossRef]
- A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012). [CrossRef]
- P. Poggiolini, G. Bosco, A. Carena, V. Curri, F. Forghieri, “A detailed analytical derivation of the GN model of non-linear interference in coherent optical transmission systems,” July2, 2012[Online]. Available: arXiv:1209.0394v12 [physics.optics].
- P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. of Light-wave Technol. 30, 3857–3879, (2012). [CrossRef]
- A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. Tapia Taiba, F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC, 2010, Paper P4.07.
- E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Proc. ECOC, 2011, Paper We.7.B.2.
- P. Johannisson, M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” IEEE J. of Lightwave Technol. 31, 1273–1282 (2013). [CrossRef]
- A. Bononi, P. Serena, “An alternative derivation of Johannissons regular perturbation model,” July19, 2012[Online]. Available: arXiv:1207.4729v1 [physics.optics].
- G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24, 1230–1232, (2012) [CrossRef]
- A. Bononi, N. Rossi, P. Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” Proceeding of the European Conf. on Opt. Comm. (ECOC), Paper We.2.C.4, Amsterdam (2012)
- S. J. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photon. Technol. Lett. 25, 961–964, (2013) [CrossRef]
- A. Mecozzi, R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30, 2011–2024 (2012). [CrossRef]
- X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach, S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nature Photonics 7, 560–568(2013). [CrossRef]
- R. Dar, M. Shtaif, M. Feder, “New bounds on the capacity of fiber-optics communications,” arXiv:1305.1762v1 [physics.optics].
- R. Dar, M. Shtaif, M. Feder, “Improved bounds on the nonlinear fiber-channel capacity,” in Proc. ECOC, 2013, paper P.4.16.
- S. Kumar, D. Yang, “Second-order theory for self-phase modulation andcross-phase modulation in optical fibers,” J. of Lightwave Technol. 23, 2073–2080, (2005) [CrossRef]
- We have assumed a fixed position independent dispersion parameter β″. The generalization to position dependent dispersion is straightforward, but we avoid it as it considerably complicates the notation.
- Although terms involving X0,k,mwith m≠ k also contribute to phase noise, they are much smaller than the terms proportional to X0,m,m(particularly in the limit of large dispersion) and we omit them from the analysis presented in this paper.
- X. Zhou, “Hardware efficient carrier recovery algorithms for single-carrier QAM systems,” in Advanced Photonics Congress, OSA Technical Digest (online)(Optical Society of America, 2012), paper SpTu3A.1. [CrossRef]
- A. Tolmachev, I. Tselniker, M. Meltsin, I. Sigron, M. Nazarathy, “Efficient multiplier-free fpga demonstration of polar-domain multi-symbol-delay-detector (MSDD) for high performance phase recovery of 16-QAM,” Proceedings of Optical Fiber Comm. Conf. OFC 2012, Paper OMC8
- A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13(1994). [CrossRef]
- R.-J. Essiambre, G. Kramer, P.J. Winzer, G.J. Foschini, B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701, (2010). [CrossRef]
- S. Asmussen, P.W. Glynn, Stochastic Simulation, Algorithms and Analysis (Springer, 2007).
- The numerical curves in Figs. 3 and 4 reproduce the plots reported in [20], but with larger statsitics.
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