429 research outputs found
On the accuracy of phase-type approximations of heavy-tailed risk models
Numerical evaluation of ruin probabilities in the classical risk model is an
important problem. If claim sizes are heavy-tailed, then such evaluations are
challenging. To overcome this, an attractive way is to approximate the claim
sizes with a phase-type distribution. What is not clear though is how many
phases are enough in order to achieve a specific accuracy in the approximation
of the ruin probability. The goals of this paper are to investigate the number
of phases required so that we can achieve a pre-specified accuracy for the ruin
probability and to provide error bounds. Also, in the special case of a
completely monotone claim size distribution we develop an algorithm to estimate
the ruin probability by approximating the excess claim size distribution with a
hyperexponential one. Finally, we compare our approximation with the heavy
traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference
Corrected phase-type approximations of heavy-tailed queueing models in a Markovian environment
Significant correlations between arrivals of load-generating events make the
numerical evaluation of the workload of a system a challenging problem. In this
paper, we construct highly accurate approximations of the workload distribution
of the MAP/G/1 queue that capture the tail behavior of the exact workload
distribution and provide a bounded relative error. Motivated by statistical
analysis, we consider the service times as a mixture of a phase-type and a
heavy-tailed distribution. With the aid of perturbation analysis, we derive our
approximations as a sum of the workload distribution of the MAP/PH/1 queue and
a heavy-tailed component that depends on the perturbation parameter. We refer
to our approximations as corrected phase-type approximations, and we exhibit
their performance with a numerical study.Comment: Received the Marcel Neuts Student Paper Award at the 8th
International Conference on Matrix Analytic Methods in Stochastic Models 201
Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times
We investigate an insurance risk model that consists of two reserves which
receive income at fixed rates. Claims are being requested at random epochs from
each reserve and the interclaim times are generally distributed. The two
reserves are coupled in the sense that at a claim arrival epoch, claims are
being requested from both reserves and the amounts requested are correlated. In
addition, the claim amounts are correlated with the time elapsed since the
previous claim arrival. We focus on the probability that this bivariate reserve
process survives indefinitely. The infinite- horizon survival problem is shown
to be related to the problem of determining the equilibrium distribution of a
random walk with vector-valued increments with reflecting boundary. This
reflected random walk is actually the waiting time process in a queueing system
dual to the bivariate ruin process. Under assumptions on the arrival process
and the claim amounts, and using Wiener-Hopf factor- ization with one
parameter, we explicitly determine the Laplace-Stieltjes transform of the
survival function, c.q., the two-dimensional equilibrium waiting time
distribution. Finally, the bivariate transforms are evaluated for some
examples, including for proportional reinsurance, and the bivariate ruin
functions are numerically calculated using an efficient inversion scheme.Comment: 24 pages, 6 figure
Queues and risk models with simultaneous arrivals
We focus on a particular connection between queueing and risk models in a
multi-dimensional setting. We first consider the joint workload process in a
queueing model with parallel queues and simultaneous arrivals at the queues.
For the case that the service times are ordered (from largest in the first
queue to smallest in the last queue) we obtain the Laplace-Stieltjes transform
of the joint stationary workload distribution. Using a multivariate duality
argument between queueing and risk models, this also gives the Laplace
transform of the survival probability of all books in a multivariate risk model
with simultaneous claim arrivals and the same ordering between claim sizes.
Other features of the paper include a stochastic decomposition result for the
workload vector, and an outline how the two-dimensional risk model with a
general two-dimensional claim size distribution (hence without ordering of
claim sizes) is related to a known Riemann boundary value problem
On exceedance times for some processes with dependent increments
Let be a random walk with a negative drift and i.i.d.
increments with heavy-tailed distribution and let be its
supremum. Asmussen & Kl{\"u}ppelberg (1996) considered the behavior of the
random walk given that , for large, and obtained a limit theorem, as
, for the distribution of the quadruple that includes the time
\rtreg=\rtreg(x) to exceed level , position Z_{\rtreg} at this time,
position Z_{\rtreg-1} at the prior time, and the trajectory up to it (similar
results were obtained for the Cram\'er-Lundberg insurance risk process). We
obtain here several extensions of this result to various regenerative-type
models and, in particular, to the case of a random walk with dependent
increments. Particular attention is given to describing the limiting
conditional behavior of . The class of models include Markov-modulated
models as particular cases. We also study fluid models, the Bj{\"o}rk-Grandell
risk process, give examples where the order of is genuinely different
from the random walk case, and discuss which growth rates are possible. Our
proofs are purely probabilistic and are based on results and ideas from
Asmussen, Schmidli & Schmidt (1999), Foss & Zachary (2002), and Foss,
Konstantopoulos & Zachary (2007).Comment: 17 page
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
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