On October 8th, 2018, Nadhir Ben Rached successfully defended
his PhD thesis entitled "Rare Events Simulations with Applications to
the Performance Evaluation of Wireless Communication Systems. "
Committee Members:
Dr. Gerardo Rubino, INRIA, France
Dr. Abla Kamoun, KAUST
Prof. Raphael Huser, KAUST
Prof Mohamed Slim Alouini, KAUST, Supervisor
Prof. Raul Tempone, KAUST - Co-supervisor
Abstract:
The probability that a sum of random variables
(RVs) exceeds (respectively falls below) a given threshold, is often
encountered in the performance analysis of wireless communication
systems. Generally, a closed-form expression of the sum distribution
does not exist and a naive Monte Carlo (MC) simulation is
computationally expensive when dealing with rare events. An alternative
approach is represented by the use of variance reduction techniques,
known for their efficiency in requiring less computations for achieving
the same accuracy requirement. For the right-tail region, we develop a
unified hazard rate twisting importance sampling (IS) technique that
presents the advantage of being logarithmic efficient for arbitrary
distributions under the independence assumption. A further improvement
of this technique is then developed wherein the twisting is applied only
to the components having more impacts on the probability of interest
than others. Another challenging problem is when the components are
correlated and distributed according to the Log-normal distribution. In
this setting, we develop a generalized hybrid IS scheme based on a mean
shifting and covariance matrix scaling techniques and we prove that the
logarithmic efficiency holds again for two particular instances. We also
propose two unified IS approaches to estimate the left-tail of sums of
independent positive RVs. The first applies to arbitrary distributions
and enjoys the logarithmic efficiency criterion, whereas the second
satisfies the bounded relative error criterion under a mild assumption
but is only applicable to the case of independent and identically
distributed RVs. The left-tail of correlated Log-normal variates is also
considered. In fact, we construct an estimator combining an existing
mean shifting IS approach with a control variate technique and prove
that it possess the asymptotically vanishing relative error property. A
further interesting problem is the left-tail estimation of sums of
ordered RVs. Two estimators are presented. The first is based on IS and
achieves the bounded relative error under a mild assumption. The second
is based on conditional MC approach and achieves the bounded relative
error property for the Generalized Gamma case and the logarithmic
efficiency for the Log-normal case.
Biography:
Nadhir Ben Rached was born in Nabeul, Tunisia. He
received the Diplôme d’Ingénieur degree from the École Polytechnique de
Tunisie, La Marsa, Tunisia, in 2012 and the M.S. degree in Applied
Mathematics and Computational Science from King Abdullah University of
Science and Technology, Thuwal, Saudi Arabia, in 2013, where he is
currently working toward the Ph.D degree in Applied Mathematics and
Computational Science. His current research interests include rare event
simulation algorithms for the accurate performance analysis of wireless
communication systems.