Abstract
In reliability theory and network performance analysis a relevant role is played by the time needed to reach a given threshold, known in probability theory as hitting time. Although such issue has been widely investigated, closed-form results are available only for independent increments of the input process. Hence, in this paper we focus on the estimation of the upper tail of the hitting time distribution for general Gaussian processes by means of discrete-event simulation. Indeed, Gaussian processes often arise as a powerful modelling tool in many real-life systems and suitable ad-hoc techniques have developed for their analysis and simulation. Since the event of interest becomes rare as the threshold increases, a variant of Conditional Monte Carlo, based on the bridge process, is introduced and the explicit expression of the estimator is derived. Finally, simulation results highlight the unbiasedness and effectiveness (in terms of relative error) of the proposed approach.
* The study was carried out under state order to the Karelian Research Centre of the Russian Academy of Sciences (Institute of Applied Mathematical Research KarRC RAS) and supported by the Russian Foundation for Basic Research, projects 18-07-00147, 18-07-00187, 19-07-00303 as well as by the University of Pisa under the PRA 2018–2019 Research Project “CONCEPT – COmmunication and Networking for vehicular CybEr-Physical sysTems”.
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References
Arutkin, M., Walter, B., Wiese, K.J.: Extreme events for fractional brownian motion with drift: theory and numerical validation. Phys. Rev. E. 102, 022102 (2020). https://doi.org/10.1103/PhysRevE.102.022102
Borodin, A.N., Salminen, P.: Handbook of Brownian Motion - Facts and Formulae. Birkhauser, Basel (2002)
Caglar, M., Vardar, C.: Distribution of maximum loss of fractional brownian motion with drift. Stat. Prob. Lett. 83, 2729–2734 (2013)
Chetvertakova, E.S., Chimitova, E.V.: The Wiener degradation model in reliability analysis. In: 2016 11th International Forum on Strategic Technology (IFOST), pp. 488–490 (2016)
Dong, Q., Cui, L.: A study on stochastic degradation process models under different types of failure thresholds. Reliab. Eng. Syst. Saf. 181, 202–212 (2019). https://doi.org/10.1016/j.ress.2018.10.002
Floyd, S., Jacobson, V.: Random early detection gateways for congestion avoidance. IEEE/ACM Trans. Netw. 1(4), 397–413 (1993)
Gao, H., Cui, L., Kong, D.: Reliability analysis for a Wiener degradation process model under changing failure thresholds. Reliab. Eng. Syst. Saf. 171, 1–8 (2018). https://doi.org/10.1016/j.ress.2017.11.006
Giordano, S., Gubinelli, M., Pagano, M.: Bridge Monte-Carlo: a novel approach to rare events of Gaussian processes. In: Proceedings of the 5th St.Petersburg Workshop on Simulation, pp. 281–286. St. Petersburg, Russia (2005)
Giordano, S., Gubinelli, M., Pagano, M.: Rare events of gaussian processes: a performance comparison between bridge monte-carlo and importance sampling. In: Koucheryavy, Y., Harju, J., Sayenko, A. (eds.) NEW2AN 2007. LNCS, vol. 4712, pp. 269–280. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74833-5_23
Glasserman, P., Wang, Y.: Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Prob. 7(3), 731–746 (1997)
Heidelberger, P.: Fast simulation of rare events in queueing and reliability models. ACM Trans. Model. Comput. Simul. 5(1), 43–85 (1995)
Kulkarni, V., Rolski, T.: Fluid model driven by an Ornstein-Uhlenbeck process. Prob. Eng. Inform. Sci. 8, 403–417 (1994)
Lukashenko, O.V., Morozov, E.V., Pagano, M.: On the efficiency of Bridge Monte-Carlo estimator. Inform. Appl. 11(2), 16–24 (2017)
Lukashenko, O.V., Morozov, E.V., Pagano, M.: A Gaussian approximation of the distributed computing process. Inform. Appl. 13(2), 109–116 (2019)
Lukashenko, O., Morozov, E., Pagano, M.: On the use of a bridge process in a conditional monte carlo simulation of gaussian queues. In: Dudin, A., Gortsev, A., Nazarov, A., Yakupov, R. (eds.) Information Technologies and Mathematical Modelling - Queueing Theory and Applications, pp. 207–220. Springer International Publishing, Cham (2016)
Michna, Z.: On tail probabilities and first passage times for fractional brownian motion. Math. Methods Oper. Res. 49, 335–354 (1999)
Norros, I.: A storage model with self-similar input. Queueing Syst. 16, 387–396 (1994)
Taqqu, M.S., Willinger, W., Sherman, R.: Proof of a fundamental result in self-similar traffic modeling. Comput. Commun. Rev. 27, 5–23 (1997)
Walter, B., Wiese, K.J.: Sampling first-passage times of fractional brownian motion using adaptive bisections. Phys. Rev. E. 101, 043312 (2020). https://doi.org/10.1103/PhysRevE.101.043312
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Lukashenko, O., Pagano, M. (2020). Rare-Event Simulation for the Hitting Time of Gaussian Processes. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds) Distributed Computer and Communication Networks. DCCN 2020. Lecture Notes in Computer Science(), vol 12563. Springer, Cham. https://doi.org/10.1007/978-3-030-66471-8_45
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