Analysis of a splitting estimator for rare event probabilities in Jackson networks

Abstract

We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in [8] that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffices to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n 2β V +1 ) function evaluations suffice to achieve a given relative precision, where βV is the number of bottleneck stations in the subset of stations under consideration in the network. This is the first rigorous analysis that favorably compares splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n d ) variables.

Randomized Dimension Reduction for Monte Carlo Simulations

“…Glasserman, Heidelberger, Shahabuddin and Zajic (1999) analyse the performance of multilevel splitting techniques for rare event estimation and give, under certain conditions, the optimal degree of splitting as the probability of the event goes to 0. Multilevel splitting methods have had many applications, such as the estimation of network reliability (Botev, L'Ecuyer, Rubino, Simard and Tuffin 2013) and of rare events in Jackson networks (Blanchet, Leder and Shi 2011). Multilevel splitting techniques for rare event simulation with finite time constraints are analysed in (Jiang and Fu 2017).…”

Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian motions

“…In [8], Dean and Dupuis showed how one can use a large deviation principle to ensure that the splitting algorithm is stable and efficiently estimates the rare event of interest. There have been several works (e.g., [8,9,2]) that have looked at rare event simulations using particle methods, but to our knowledge, our work is the first to study rare event simulations for SRBMs using any method.…”