Proceedings of the 2007 Winter Simulation Conference S. G. Henderson, B. Biller, M.-H. Hsieh, J. Shortle, J. D. Tew, and R. R. Barton, eds. EFFICIENT SUBOPTIMAL RARE-EVENT SIMULATION Xiaowei Zhang Department of Management Science and Engineering Stanford University Stanford, C.A. 94305, U.S.A. Jose Blanchet Department of Statistics Harvard University Cambridge, M.A. 02138, U.S.A Peter W. Glynn Department of Management Science and Engineering Stanford University Stanford, C.A. 94305, U.S.A. ABSTRACT Much of the rare-event simulation literature is concerned with the development of asymptotically optimal algorithms. Because of the difficulties associated with applying these ideas to complex models, this paper focuses on sub-optimal procedures that can be shown to be much more efficient than conventional crude Monte Carlo. We provide two such examples, one based on “repeated acceptance/rejection” as a mean of computing tail probabilities for hitting time random variables and the other based on filtered conditional Monte Carlo. 1 INTRODUCTION The rare-event simulation problem is concerned with using simulation to compute α = P(A), where A is a “rare-event” (and hence has probability P(A) close to zero). It is well known that if α is computed via crude Monte Carlo (i.e., by estimating α via the proportion of independent simulations on which the event A occurs), the number of trials n required to estimate α to a given relative precision scales is in pro- portion to 1/P(A). As a consequence, crude Monte Carlo (CMC) is a highly inefficient algorithm for computing α when P(A) is small. Much of the rare-event simulation liter- ature is concerned with the development of asymptotically optimal algorithms. Because of the difficulties associated with applying these ideas to complex models, this paper focuses on sub-optimal procedures that can be shown to be much more efficient than conventional crude Monte Carlo. We provide two such examples, one based on “repeated acceptance/rejection” as a mean of computing tail probabil- ities for hitting time random variables and the other based on filtered conditional Monte Carlo. It follows that for very rare events (say, of order 10 -4 or smaller), there is great interest (both practically speaking and academically) in using modified simulation algorithms capable of computing α more efficiently (i.e., using a so- called “efficiency improvement technique”). Accordingly, there is now a substantial literature on such efficient rare- event simulation algorithms; see, for example, Bucklew (2004) and Juneja and Shahabuddin (2007). Given a family of problem instances (P(A θ ) : θ ∈ Λ), let Z(θ ) be a random variable having mean equal to P(A θ ) for each θ ∈ Λ. The family of r.v.’s (Z(θ ) : θ ∈ Λ) is said to have bounded relative variance (BRV) if sup θ ∈Λ varZ(θ ) P(A θ ) 2 < ∞. Bounded relative variance implies, via Chebyshev’s inequality, that the number of observations n required to compute P(A θ ) to a given relative precision is bounded in θ ∈ Λ. In particular, this implies that if P(A θ ) ↓ 0 as θ → θ 0 , then EZ 2 (θ )= O(P(A θ ) 2 ) as θ → θ 0 . Because the Cauchy-Schwarz inequality implies that EZ 2 (θ ) ≥ P(A θ ) 2 , it follows that EZ 2 (θ ) is within a constant multiple, uniformly in θ ∈ Λ, of the smallest possible second moment. Hence, the family (EZ 2 (θ ) : θ ∈ Λ) is within a constant multiple of optimality. An extensive theory of asymptotic optimality based on the concept of BRV (and a weaker notion known as “logarithmic optimality”) has developed in response to rare- event simulation problems in several applications domains. In order to guarantee asymptotic optimality, one clearly 389 1-4244-1306-0/07/$25.00 ©2007 IEEE