8 research outputs found
How to estimate a cumulative process’s rate-function
Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate-function J(·,·) and whose cumulative process satisfies the LDP with rate-function I(·). Under mixing conditions, an LDP for estimates of I
constructed by transforming an estimate of J is proved. For the case of cumulative renewal processes it is demonstrated that this approach is favorable to a more direct method as it ensures the laws of the estimates converge weakly to a Dirac measure at I
How to estimate a cumulative process’s rate-function
Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate-function J(·,·) and whose cumulative process satisfies the LDP with rate-function I(·). Under mixing conditions, an LDP for estimates of I
constructed by transforming an estimate of J is proved. For the case of cumulative renewal processes it is demonstrated that this approach is favorable to a more direct method as it ensures the laws of the estimates converge weakly to a Dirac measure at I
Robust measurement-based buffer overflow probability estimators for QoS provisioning and traffic anomaly prediction applicationm
Suitable estimators for a class of Large Deviation approximations of rare
event probabilities based on sample realizations of random processes have been
proposed in our earlier work. These estimators are expressed as non-linear
multi-dimensional optimization problems of a special structure. In this paper,
we develop an algorithm to solve these optimization problems very efficiently
based on their characteristic structure. After discussing the nature of the
objective function and constraint set and their peculiarities, we provide a
formal proof that the developed algorithm is guaranteed to always converge. The
existence of efficient and provably convergent algorithms for solving these
problems is a prerequisite for using the proposed estimators in real time
problems such as call admission control, adaptive modulation and coding with
QoS constraints, and traffic anomaly detection in high data rate communication
networks
Robust measurement-based buffer overflow probability estimators for QoS provisioning and traffic anomaly prediction applications
Suitable estimators for a class of Large Deviation approximations of rare event probabilities based on sample realizations of random processes have been proposed in our earlier work. These estimators are expressed as non-linear multi-dimensional optimization problems of a special structure. In this paper, we develop an algorithm to solve these optimization problems very efficiently based on their characteristic structure. After discussing the nature of the objective function and constraint set and their peculiarities, we provide a formal proof that the developed algorithm is guaranteed to always converge. The existence of efficient and provably convergent algorithms for solving these problems is a prerequisite for using the proposed estimators in real time problems such as call admission control, adaptive modulation and coding with QoS constraints, and traffic anomaly detection in high data rate communication networks
Most likely paths to error when estimating the mean of a reflected random walk
It is known that simulation of the mean position of a Reflected Random Walk
(RRW) exhibits non-standard behavior, even for light-tailed increment
distributions with negative drift. The Large Deviation Principle (LDP) holds
for deviations below the mean, but for deviations at the usual speed above the
mean the rate function is null. This paper takes a deeper look at this
phenomenon. Conditional on a large sample mean, a complete sample path LDP
analysis is obtained. Let denote the rate function for the one dimensional
increment process. If is coercive, then given a large simulated mean
position, under general conditions our results imply that the most likely
asymptotic behavior, , of the paths is
to be zero apart from on an interval and to satisfy the
functional equation \begin{align*} \nabla
I\left(\ddt\psi^*(t)\right)=\lambda^*(T_1-t) \quad \text{whenever } \psi(t)\neq
0. \end{align*} If is non-coercive, a similar, but slightly more involved,
result holds.
These results prove, in broad generality, that Monte Carlo estimates of the
steady-state mean position of a RRW have a high likelihood of over-estimation.
This has serious implications for the performance evaluation of queueing
systems by simulation techniques where steady state expected queue-length and
waiting time are key performance metrics. The results show that na\"ive
estimates of these quantities from simulation are highly likely to be
conservative.Comment: 23 pages, 8 figure