Abstract
We consider a linear stochastic fluid network under Markov modulation, with a focus on the probability that the joint storage level attains a value in a rare set at a given point in time. The main objective is to develop efficient importance sampling algorithms with provable performance guarantees. For linear stochastic fluid networks without modulation, we prove that the number of runs needed (so as to obtain an estimate with a given precision) increases polynomially (whereas the probability under consideration decays essentially exponentially); for networks operating in the slow modulation regime, our algorithm is asymptotically efficient. Our techniques are in the tradition of the rare-event simulation procedures that were developed for the sample-mean of i.i.d. one-dimensional light-tailed random variables, and intensively use the idea of exponential twisting. In passing, we also point out how to set up a recursion to evaluate the (transient and stationary) moments of the joint storage level in Markov-modulated linear stochastic fluid networks.
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References
Asmussen S, Glynn P (2007) Stochastic simulation. Springer, New York
Asmussen S, Kortschak D (2015) Error rates and improved algorithms for rare event simulation with heavy Weibull tails. Methodol Comput Appl Probab 17:441–461
Asmussen S, Nielsen H (1995) Ruin probabilities via local adjustment coefficients. J Appl Probab 33:736–755
Bahadur R, Rao RR (1960) On deviations of the sample mean. Ann Math Stat 31:1015–1027
Blanchet J, Leder K, Glynn P (2008) Strongly efficient algorithms for light-tailed random walks: an old folk song sung to a faster new tune. In: L’Ecuyer P, Owen A (eds) Proceedings of the Eighth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC 2008). Springer, Berlin, pp 227–248
Blanchet J, Mandjes M (2009) Rare event simulation for queues. In: Rubino G, Tuffin B (eds) Rare event simulation using Monte Carlo methods. Wiley, Chichester, pp 87–124
Blom J, De Turck K, Mandjes M (2017) Refined large deviations asymptotics for Markov-modulated infinite-server systems. Euro J Oper Res 259:1036–1044
Blom J, Mandjes M (2013) A large-deviations analysis of Markov-modulated infinite-server queues. Oper Res Lett 41:220–225
Boxma O, Cahen E, Koops D, Mandjes M (2018) Linear networks: rare-event simulation and Markov modulation. arXiv:1705.10273
Budhiraja A, Nyquist P (2015) Large deviations for multidimensional state-dependent shot-noise processes. J Appl Probab 52:1097–1114
Cahen E, Mandjes M, Zwart B (2017) Rare event analysis and efficient simulation for a multi-dimensional ruin problem. Probab Eng Inform Sci 31:265–283
Chaganthy N, Sethuraman J (1996) Multidimensional strong large deviation theorems. J Stat Plan Inference 55:265–280
Dembo A, Zeitouni O (1998) Large deviations techniques and applications, 2nd ed. Springer, New York
Ganesh A, Macci C, Torrisi G (2007) A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling. Queueing Syst 55:83–94
Glasserman P, Juneja S (2008) Uniformly efficient importance sampling for the tail distribution of sums of random variables. Math Oper Res 33:36–50
Huang G, Jansen HM, Mandjes M, Spreij P, De Turck K (2016) Markov-modulated Ornstein-Uhlenbeck processes. Adv Appl Probab 48:235–254
Huang G, Mandjes M, Spreij P (2016) Large deviations for Markov-modulated diffusion processes with rapid switching. Stoch Process Appl 126:1785–1818
Juneja S, Shahabuddin P, Nelson B (2006) Rare event simulation techniques: an introduction and recent advances. In: Henderson S (ed) Handbook in operations research and management sciences, volume 13: Simulation, pp 291–350
Kella O, Stadje W (2002) Markov modulated linear fluid networks with Markov additive input. J Appl Probab 39:413–420
Kella O, Whitt W (1999) Linear stochastic fluid networks. J Appl Probab 36:244–260
Kuhn J, Mandjes M, Taimre T (2017) Exact asymptotics of sample-mean related rare-event probabilities. Probability in the Engineering and Informational Sciences, to appear
Magnus J, Neudecker H (1979) The commutation matrix: some properties and applications. Ann Stat 7:381–394
Rabehasaina L (2006) Moments of a Markov-modulated irreducible network of fluid queues. J Appl Probab 43:510–522
Sezer A (2009) Importance sampling for a Markov modulated queuing network. Stoch Process Appl 119:491–517
Acknowledgments
The research of O. Boxma, D. Koops and M. Mandjes was partly funded by the NWO Gravitation Project Networks, Grant Number 024.002.003. The research of O. Boxma was also partly funded by the Belgian Government, via the IAP Bestcom project. The research of E. Cahen was funded by an NWO grant, Grant Number 613.001.352.
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Appendix A
Appendix A
We here point out how (6) can be established; the line of reasoning is precisely the same as in the derivation of (5) in (Dembo and Zeitouni 1998)(Thm. 3.7.4). First write
which, with \(Z_{n}:=(Y_{n}(t)-na)/\sqrt {n}\), equals
Observe that \({\mathbb E}_{\mathbb Q}\,Y_{n}=na\), due to the very choice of \({\mathbb Q}\). This entails that Zn converges in distribution to a centered Normal random variable; as can be verified, the corresponding variance is τ (where τ is defined in Eq. 5). Using the Berry-Esseen-based justification presented in (Dembo and Zeitouni 1998)(page 111), we conclude that, as n →∞,
Completing the square, the right-hand side of the previous display equals, with \({\mathscr N}(\textsc {m},\textsc {v})\) a normal random variable with mean m and variance v,
Now we use the standard equivalence (as x →∞)
to obtain
Combining the above, we derive the claim:
We now proceed with the computations underlying (12). To this end, first observe that
As a consequence, in line with the above computation for the one-dimensional case,
It was proven in (Chaganthy and Sethuraman 1996) (Thm. 3.4) that
while at the same time
This immediately leads to Eq. 12.
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Boxma, O.J., Cahen, E.J., Koops, D. et al. Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation. Methodol Comput Appl Probab 21, 125–153 (2019). https://doi.org/10.1007/s11009-018-9644-1
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DOI: https://doi.org/10.1007/s11009-018-9644-1