Benzion Boukai

Bio: Benzion Boukai is an academic researcher from Purdue University. The author has contributed to research in topics: Geometric Brownian motion & Brownian motion. The author has an hindex of 2, co-authored 2 publications receiving 28 citations.

An explicit expression for the distribution of the supremum of brownian motion with a change point

Abstract: The objective of this paper is to derive explicit expressions to distributions of the supremum of Brownian motion processes having a change-point. Such processes are characterized by a drift parameter which is subjected to a change over time. For that purpose, we use the distribution of the supremum of Brownian bridge process, via conditioning arguments. Several different cases are considered and the resulting distributions are illustrated through their p.d.f.'s.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

First-exit times of an inverse Gaussian process

Abstract: The first-exit time process of an inverse Gaussian Levy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail...

Change-Point Problems: Bibliography and Review

Abstract: Five types of change-point problems concerning change in mean, variance, slope, hazard rate, and space-time distribution are briefly reviewed and a list of comprehensive bibliography is provided. Directions for future studies are discussed.

An explicit expression for the distribution of the supremum of brownian motion with a change point

Abstract: The objective of this paper is to derive explicit expressions to distributions of the supremum of Brownian motion processes having a change-point. Such processes are characterized by a drift parameter which is subjected to a change over time. For that purpose, we use the distribution of the supremum of Brownian bridge process, via conditioning arguments. Several different cases are considered and the resulting distributions are illustrated through their p.d.f.'s.

On the steady-state probability of delay and large negative deviations for the $GI/GI/n$ queue in the Halfin-Whitt regime

Abstract: We consider the FCFS $GI/GI/n$ queue in the Halfin-Whitt heavy traffic regime, and prove bounds for the steady-state probability of delay (s.s.p.d.) for generally distributed processing times. We prove that there exist $\epsilon_1, \epsilon_2 > 0$, depending on the first three moments of the inter-arrival and processing time distributions, such that the s.s.p.d. is bounded from above by $\exp\big(-\epsilon_1 B^2\big)$ as the associated excess parameter $B \rightarrow \infty$; and by $1 - \epsilon_2 B$ as $B \rightarrow 0$. We also prove that the tail of the steady-state number of idle servers has a Gaussian decay. We provide explicit bounds in all cases, in terms of the first three moments of the inter-arrival and service distributions, and use known results to show that our bounds correctly capture various qualitative scalings. \\\indent Our main proof technique is the derivation of new stochastic comparison bounds for the FCFS $GI/GI/n$ queue, which are of a structural nature, hold for all $n$ and times $t$, and significantly generalize the work of \citet{GG.10c} (e.g. by providing bounds for the queue length to exceed any given level, as opposed to any given level strictly greater than the number of servers as acheived in \citet{GG.10c}). Our results do not follow from simple comparison arguments to e.g. infinite-server systems or loss models, which would in all cases provide bounds in the opposite direction.