When Machine Learning Meets Importance Sampling: A More Efficient Rare Event Estimation Approach

TL;DR

This work tackles the problem of estimating the steady-state tail probability $p_\gamma = \mathbb{P}(X^1+X^2 \geq \gamma)$ in a two-node tandem queue, where standard regenerative IS suffers from path-dependent variance. It proposes a Marginal Stationary IS (MIS) framework that targets the stationary distribution of an alternative system and learns the stationary likelihood ratio $\pi/\tilde{\pi}$ via off-policy evaluation in an RKHS, implemented as a neural-network-based MLIS algorithm. The learning objective minimizes a robust RKHS-based loss to identify $\pi/\tilde{\pi}$, enabling an IS estimator that uses weights $\pi(\tilde{X})/\tilde{\pi}(\tilde{X})$ under $\tilde{\pi}$ and avoids path-dependence variance. Numerical experiments show that MLIS outperforms classical RIS, is competitive with MIS, and remains robust across different target rarities, suggesting practical gains for SLA-related tail estimates in networks. For the two-node tandem case, the exact tail probability is $\mathbb{P}(X^1+X^2\ge\gamma)=\frac{(1-\rho_1)\rho_2^{\gamma+1}-(1-\rho_2)\rho_1^{\gamma+1}}{\rho_2-\rho_1}$ with $\rho_i=\lambda/\mu_i$, illustrating the benchmark against which MLIS improvements are measured.

Abstract

Driven by applications in telecommunication networks, we explore the simulation task of estimating rare event probabilities for tandem queues in their steady state. Existing literature has recognized that importance sampling methods can be inefficient, due to the exploding variance of the path-dependent likelihood functions. To mitigate this, we introduce a new importance sampling approach that utilizes a marginal likelihood ratio on the stationary distribution, effectively avoiding the issue of excessive variance. In addition, we design a machine learning algorithm to estimate this marginal likelihood ratio using importance sampling data. Numerical experiments indicate that our algorithm outperforms the classic importance sampling methods.

Figures (2)

• Figure 1: Structure of a 2-station tandem queueing network.
• Figure 2: Two representative sample paths reaching the target event $\{X^1 + X^2 \geq \gamma\}$ from the empty state.

Theorems & Definitions (1)

• Proposition 1