18 research outputs found
Simulating Tail Probabilities in GI/GI.1 Queues and Insurance Risk Processes with Subexponentail Distributions
This paper deals with estimating small tail probabilities of thesteady-state waiting time in a GI/GI/1 queue with heavy-tailed (subexponential) service times. The problem of estimating infinite horizon ruin probabilities in insurance risk processes with heavy-tailed claims can be transformed into the same framework. It is well-known that naive simulation is ineffective for estimating small probabilities and special fast simulation techniques like importance sampling, multilevel splitting, etc., have to be used. Though there exists a vast amount of literature on the rare event simulation of queuing systems and networks with light-tailed distributions, previous fast simulation techniques for queues with subexponential service times have been confined to the M/GI/1 queue. The general approach is to use the Pollaczek-Khintchine transformation to convert the problem into that of estimating the tail distribution of a geometric sum of independent subexponential random variables. However, no such useful transformation exists when one goes from Poisson arrivals to general interarrival-time distributions. We describe and evaluate an approach that is based on directly simulating the random walk associated with the waiting-time process of the GI/GI/1 queue, using a change of measure called delayed subexponential twisting -an importance sampling idea recently developed and found useful in the context of M/GI/1 heavy-tailed simulations
Achieving Efficiency in Black Box Simulation of Distribution Tails with Self-structuring Importance Samplers
Motivated by the increasing adoption of models which facilitate greater
automation in risk management and decision-making, this paper presents a novel
Importance Sampling (IS) scheme for measuring distribution tails of objectives
modelled with enabling tools such as feature-based decision rules, mixed
integer linear programs, deep neural networks, etc. Conventional efficient IS
approaches suffer from feasibility and scalability concerns due to the need to
intricately tailor the sampler to the underlying probability distribution and
the objective. This challenge is overcome in the proposed black-box scheme by
automating the selection of an effective IS distribution with a transformation
that implicitly learns and replicates the concentration properties observed in
less rare samples. This novel approach is guided by a large deviations
principle that brings out the phenomenon of self-similarity of optimal IS
distributions. The proposed sampler is the first to attain asymptotically
optimal variance reduction across a spectrum of multivariate distributions
despite being oblivious to the underlying structure. The large deviations
principle additionally results in new distribution tail asymptotics capable of
yielding operational insights. The applicability is illustrated by considering
product distribution networks and portfolio credit risk models informed by
neural networks as examples.Comment: 51 page
Current Topics on Risk Analysis: ICRA6 and RISK2015 Conference
Peer ReviewedPostprint (published version