Proceedings of the 2013 Winter Simulation Conference R. Pasupathy, S.-H. Kim, A. Tolk, R. Hill, and M. E. Kuhl, eds. RARE EVENT SIMULATION FOR STOCHASTIC FIXED POINT EQUATIONS RELATED TO THE SMOOTHING TRANSFORMATION Jeffrey F. Collamore Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen, Denmark Anand N. Vidyashankar Department of Statistics Volgeneau School of Engineering Georgia Mason University Fairfax, VA 22030, U.S.A. Jie Xu Systems Engineering & Operations Research Department Volgeneau School of Engineering Georgia Mason University Fairfax, VA 22030, U.S.A. ABSTRACT In several applications arising in compute science, cascade theory, and other applied areas, it is of interest to evaluate the tail probabilities of non-homogeneous stochastic fixed point equations. Recently, techniques have been developed for the related linear recursions, yielding tail estimates and importance sampling methods for these recursions. However, such methods do not routinely generalize to non-homogeneous recursions. Drawing on techniques from the weighted branching process literature, we present a consistent, strongly efficient importance sampling algorithm for estimating the tail probabilities for the case of non- homogeneous recursions. 1 INTRODUCTION This paper is concerned with rare event simulation related to the non-homogeneous stochastic fixed point equations of the form V d = N ∑ i=1 A i V i + B, (1) where V ≡{ V, V i : i ≥ 1} is a collection of independent and identically distributed (i.i.d.) random variables; A ≡{A i : i ≥ 1} is a collection of non-negative random variables and B a real valued random variable, both independent of V ; and N is an integer-valued random variable, independent of V , A , and B. When B = 0, (1) is referred to as a homogeneous stochastic fixed point equation (SFPE). These SFPEs arise in a variety of examples; for instance: (i) the Quicksort algorithm, where V represents the stationary solution to the normalized key comparisons needed to sort a random permutation of length n; (ii) the Hausdorff dimension of Cantor sets; (iii) the stochastic approximation of Google’s page rank algorithm; and (iv) the study of martingale limits of Mandelbrot’s cascades and of branching random walk. 555 978-1-4799-3950-3/13/$31.00 ©2013 IEEE