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RARE EVENT ANALYSIS AND EFFICIENT SIMULATION FOR A MULTI-DIMENSIONAL RUIN PROBLEM

Published online by Cambridge University Press:  23 January 2017

Ewan Jacov Cahen ,
Michel Mandjes  and
Bert Zwart
Ewan Jacov Cahen
Affiliation:
CWI, Amsterdam, the Netherlands, E-mail: ewan.cahen@cwi.nl
Michel Mandjes
Affiliation:
CWI, Amsterdam, the Netherlands, University of Amsterdam, Amsterdam, the Netherlands E-mail: m.r.h.mandjes@uva.nl
Bert Zwart
Affiliation:
CWI, Amsterdam, the Netherlands Eindhoven University of Technology, Eindhoven, the Netherlands E-mail: bert.zwart@cwi.nl
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Abstract

This paper focuses on the evaluation of the probability that both components of a bivariate stochastic process ever simultaneously exceed some large level; a leading example is that of two Markov fluid queues driven by the same background process ever reaching the set (u, ∞)×(u, ∞), for u>0. Exact analysis being prohibitive, we resort to asymptotic techniques and efficient simulation, focusing on large values of u. The first contribution concerns various expressions for the decay rate of the probability of interest, which are valid under Gärtner–Ellis-type conditions. The second contribution is an importance-sampling-based rare-event simulation technique for the bivariate Markov modulated fluid model, which is capable of asymptotically efficiently estimating the probability of interest; the efficiency of this procedure is assessed in a series of numerical experiments.

Keywords

Large deviationsimportance samplingrare-event simulationruin probability
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 31 , Issue 3 , July 2017 , pp. 265 - 283
Copyright
Copyright © Cambridge University Press 2017 

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References

1. Biggins, J. & Sani, A., (2004). Extended Perron–Frobenius results. University of Sheffield, Department of Probability and Statistics.Google Scholar
2. Collamore, J. (1996). Hitting probabilities and large deviations. The Annals of Probability, 24(4): 2065–2078.Google Scholar
3. Duffield, N. & O'Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue, with applications. In Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge, UK: Cambridge University Press, vol. 118, pp. 363–374.Google Scholar
4. Elwalid, A. & Mitra, D. (1993). Effective bandwidth of general {M}arkovian traffic sources and admission control of high speed networks. IEEE/ACM Transactions on Networking (TON) 1(3): 329–343.Google Scholar
5. Gallier, J. (2012). Notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi diagrams and Delaunay triangulations. Book in progress, http://www.cis.upenn.edu/jean/gbooks/convexpoly.html.Google Scholar
6. Ganesh, A., O'Connell, N., & Wischik, D. (2004). Big queues. Berlin, Heidelberg: Springer.Google Scholar
7. Glynn, P. & Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. Journal of Applied Probability 31: 131–156.Google Scholar
8. Kella, O. (1993). Parallel and tandem fluid networks with dependent Lévy inputs. The Annals of Applied Probability, 3(3): 682–695.Google Scholar
9. Kesidis, G., Walrand, J., & Chang, C.-S. (1993). Effective bandwidths for multiclass {M}arkov fluids and other {ATM} sources. IEEE/ACM Transactions on Networking (TON) 1(4): 424–428.Google Scholar
10. Kosiński, K. & Mandjes, M. (2015). Logarithmic asymptotics for multidimensional extremes under nonlinear scalings. Journal of Applied Probability 52(1): 68–81.Google Scholar
11. Mandjes, M. & Ridder, A. (1995). Finding the conjugate of {M}arkov fluid processes. Probability in the Engineering and Informational Sciences 9(02): 297–315.Google Scholar
12. Rubino, G. & B, Tuffin. (2009). Rare event simulation using {M}onte {C}arlo methods. Chichester, UK: John Wiley & Sons.Google Scholar
13. Seneta, E. (1973). Non-negative matrices. London: George Allen & Unwin.Google Scholar