[Paper Review] Asymptotically Optimal Importance Sampling for Jackson Networks with a Tree Topology

Ali Devin Sezer|ArXiv.org|Aug 23, 2007

Probability and Risk ModelsDecision Sciences8 references3 citations

TL;DR

This paper develops asymptotically optimal importance sampling (IS) algorithms for rare buffer overflow events in stable open Jackson networks with a tree topology, using a subsolution approach to the Hamilton-Jacobi-Bellman (HJB) equation. It introduces a recursive algorithm to construct smooth, piecewise-affine subsolutions based on effective node utilities and rates, achieving asymptotic optimality for both shared and individual buffer structures.

ABSTRACT

Importance sampling (IS) is a variance reduction method for simulating rare events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17(4):1306- 1346, 2007) exploits connections between IS and stochastic games and optimal control problems to show how to design and analyze simple and efficient IS algorithms for various overflow events for tandem Jackson networks. The present paper uses the same approach to build asymptotically optimal IS schemes for stable open Jackson networks with a tree topology. Customers arrive at the single root of the tree. The rare overflow event we consider is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: 1) A single system-wide buffer of size $n$ shared by all nodes, 2) each node $i$ has its own buffer of size $β_i n$, $β_i \in (0,1)$.

Motivation & Objective

• To design asymptotically optimal importance sampling schemes for rare buffer overflow events in stable open Jackson networks with tree topology.
• To extend the subsolution approach to HJB equations and boundary conditions for networks beyond tandem configurations.
• To handle two buffer structures: a single system-wide buffer of size $ n $, and individual buffers of size $ \beta_i n $ at each node.
• To construct smooth subsolutions that yield provably optimal IS algorithms through effective node utilities and rates.
• To provide a recursive algorithm that generates subsolutions with heuristic interpretations as effective utilities and service rates.

Proposed method

• Uses the optimal control/subsolution framework from prior work to reduce IS design to constructing smooth subsolutions of the HJB equation and its boundary conditions.
• Employs a recursive algorithm that computes subsolutions as a smoothed minimum of affine functions, parameterized by effective gradients derived from node states.
• Defines effective gradients based on node buffer states (empty or nonempty), with associated utilities and rates that reflect system dynamics.
• Applies a weighted exponential smoothing (softmax-like) transformation to combine affine functions, ensuring smoothness and Lipschitz continuity.
• Uses a parameterized family of functions $ W^{\epsilon,\delta}(x) $ with $ \epsilon, \delta > 0 $ to control approximation accuracy and ensure subsolution properties.
• Proves that the constructed subsolutions satisfy all conditions of the general optimality theorem in [16], ensuring asymptotic optimality of the resulting IS estimator.

Experimental results

Research questions

• RQ1 Can asymptotically optimal IS be constructed for buffer overflow in tree-structured Jackson networks using the subsolution approach?
• RQ2 How can effective node utilities and rates be defined to reflect the impact of buffer states on overflow likelihood?
• RQ3 Does the recursive construction of subsolutions yield a smooth, valid subsolution satisfying the HJB equation and boundary conditions?
• RQ4 Is the resulting IS algorithm asymptotically optimal for both shared and individual buffer configurations?
• RQ5 What is the relationship between the proposed method and the more general framework in Dupuis and Wang (2010)?

Key findings

• The proposed recursive algorithm constructs smooth subsolutions of the HJB equation and its boundary conditions, which are necessary and sufficient for asymptotic optimality of IS.
• The subsolutions are of the form $ W(x) = -\delta \log \sum_{l=1}^{L} \exp\left( -\frac{1}{\delta}(2\gamma - \alpha_l \epsilon + \langle q_l, x \rangle) \right) $, ensuring smoothness and proper scaling.
• For any $ x \in \mathbb{R}_+^d $, the subsolution satisfies $ H_b(DW(x)) \geq -C_1 \exp(-\epsilon/\delta) $, proving near-optimality as $ \epsilon, \delta \to 0 $.
• The algorithm ensures that the IS estimator achieves asymptotic efficiency, with variance growing sub-exponentially relative to the rare event probability.
• Numerical results in Sections 5 and 6 confirm the practical effectiveness and low variance of the proposed IS schemes.
• The method generalizes prior results on tandem networks and provides a systematic framework applicable to arbitrary tree-structured Jackson networks.