Splitting for Rare Event Simulation: A Large Deviation Approach to Design and Analysis

Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set $B$ before another set $A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.Comment: Submitted to Stochastic Processes and their Application

Asymptotically optimal importance sampling for Jackson networks with a tree topology

This note describes an importance sampling (IS) algorithm to estimate buffer overflows of stable Jackson networks with a tree topology. Three new measures of service capacity and traffic in Jackson networks are introduced and the algorithm is defined in their terms. These measures are effective service rate, effective utilization and effective service-to-arrival ratio of a node. They depend on the nonempty/empty states of the queues of the network. For a node with a nonempty queue, the effective service rate equals the node's nominal service rate. For a node i with an empty queue, it is either a weighted sum of the effective service rates of the nodes receiving traffic directly from node i, or the nominal service rate, whichever smaller. The effective utilization is the ratio of arrival rate to the effective service rate and the effective service-to-arrival ratio is its reciprocal. The rare overflow event of interest 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 shared by all nodes, and (2) each node has its own fixed size buffer. The constructed IS algorithm is asymptotically optimal, i. e., the variance of the associated estimator decays exponentially in the buffer size at the maximum possible rate. This is proved using methods from (Dupuis et al. in Ann. Appl. Probab. 17(4): 1306-1346, 2007), which are based on a limit Hamilton-Jacobi-Bellman equation and its boundary conditions and their smooth subsolutions. Numerical examples involving networks with as many as eight nodes are provided

Alternative proof and interpretations for a recent state-dependent importance sampling scheme

Recently, a state-dependent change of measure for simulating overflows in the two-node tandem queue was proposed by Dupuis et al. (Ann. Appl. Probab. 17(4):1306–1346, 2007), together with a proof of its asymptotic optimality. In the present paper, we present an alternative, shorter and simpler proof. As a side result, we obtain interpretations for several of the quantities involved in the change of measure in terms of likelihood ratios

Efficient Rare-event Simulation for Perpetuities

We consider perpetuities of the form D = B_1 exp(Y_1) + B_2 exp(Y_1+Y_2) + ... where the Y_j's and B_j's might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the Y_j's satisfy the so-called Cramer condition with associated root theta_{ast} in (0,infty) and that the tails of the B_j's are appropriately behaved so that D is regularly varying with index theta_{ast}. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the Y_j's according to theta_{ast} fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient

Importance sampling for non-Markovian tandem queues using subsolutions

In this paper, we use importance sampling simulation to estimate the probability that the number of customers in a d-node GI|GI|1 tandem queue reaches some high level N in a busy cycle of the system. We present a state-dependent change of measure for a d-node GI|GI|1 tandem queue based on the subsolution approach, and we prove, under a mild conjecture, that this state-dependent change of measure gives an asymptotically efficient estimator for the probability of interest when all supports are bounded

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers