# THE CAP 13TH ANNUAL APPLIED PROBABILITY DAY

FRIDAY MAY 12
DAVIS AUDITORIUM, COLUMBIA UNIVERSITY

Schedule of Events

• 9:15 - 10:00 Poisson Process Approximation: from Palm theory to Stein's method

Louis H Y Chen
Institute for Mathematical Sciences
National University of Singapore

Poisson process approximation using Stein's method has been successfully developed by Barbour, Brown, Xia and others since 1988. The key idea is to convert the Stein equation to one involving the generator of an immigration-death process whose equilibrium distribution is the approximating Poisson process, solve the equation in terms of the immigration-death process and then obtain sharp bounds on the solution and its smoothness by using coupling. This approach is known as the probabilistic approach of Barbour.

In this talk, the probabilistic approach of Barbour is used but the framework of Stein's method is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence of point processes. A Wasserstein pseudo-metric is also defined and applied to certain point processes which can be viewed as locally dependent by enlarging the carrier space.

Poisson process approximation theorems are proved for locally dependent point processes as well as for dependent superposition of point processes. The theorems are applied to Matern hard-core processes, words in DNA and superposition of renewal processes.

This talk is based on joint work with Aihua Xia.

• 10:00 - 10:45 Stochastic Batch Scheduling and the "Smallest Variance First" Rule

Michael Pinedo
Stern School of Business
New York University

Consider a single machine that can process multiple jobs in batch mode. We have n jobs and the processing time of job j is a random variable X_j with distribution F_j. Up to b jobs can be processed simultaneously by the machine. The jobs in a batch all have to start at the same time and the batch is completed when all jobs have finished their processing (i.e., at the maximum of the processing times of the jobs in that batch). We are interested in two objective functions, namely the minimization of the expected makespan and the minimization of the total expected completion time. We first show that under certain fairly general conditions the minimization of the expected makespan is equivalent to specific deterministic combinatorial problems, namely the Weighted Matching problem and the Set Partitioning problem. We then consider the case when all jobs have the same mean processing time, but different variances. We show that for certain special classes of processing time distributions the "Smallest Variance First" rule minimizes the expected makespan as well as the total expected completion time. In our conclusions we present various general rules that are suitable for the minimization of the expected makespan and the total expected completion time in batch scheduling.

• 11:15 - 12:00 Stochastic Modeling in Nanoscale Biophysics

Samuel Kou
Harvard University

Recent advances in nanotechnology allow scientists to follow a biological process on the individual molecule basis. These advances also raise many challenging stochastic modeling problems, because the experimental capability of zooming in on single molecules reveal that many classical models derived from oversimplified assumptions are no longer valid. One such phenomenon that we will focus in the talk is that of subdiffusion, which much departs from the classical Brownian diffusion theory. By introducing fractional Gaussian noise (i.e. the derivative of fractional Brownian motion) into the generalized Langevin equation, we propose a model to describe subdiffusion. In addition to analytical tractability and clear physical meaning, this model is capable of explaining the experimentally observed conformational fluctuation in enzyme reactions. Excellent agreement between the model prediction and the single-molecule experimental data is seen.

• 2:00 - 2:45 On Ruin Probability for a Risk Process with Phase-type Claims and Inter-arrival Times Perturbed by a Levy Process with No Negative Jumps

Esther Frostig
University of Haifa, Israel

We study a risk process where the claim size and the inter-arrival times are phase-type distributed. The risk process is perturbed by a Levy process without negative jumps. We show that the ruin probability, and the distribution of deficit at ruin, are the same as in an unperturbed risk model with general inter-arrival times and phase type claim size, where the inter-arrival times and the claims are dependent. The model is analyzed via the dual queueing system. We show that the dual queueing system is a Markov arrival process. queueing system.

• 2:45 - 3:30 A Levy Process Reflected at a Poisson Age Process

Offer Kella
Hebrew University of Jerusalem

We consider a Levy process with no negative jumps, reflected at a stochastic boundary which is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution which is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other has the stationary distribution of a certain clearing process. The latter is itself distributed like an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail. This talk is based on joint work with Onno Boxma and Michel Mandjes.

• 4:00 - 4:45 Sampling and Estimation from Heavy Tailed Distributions in the Internet

Nick Duffield
AT&T

Internet service providers commonly collect usage data in the form of flow records that summarize sets of related packets passing through routers. Speed and bandwidth constraints in the measurement and analysis infrastructure necessitate that the flow records be sampled to reduce data volumes and increase query speed. A relatively small proportion of these flow records represent a large proportion of the traffic. This talk reviews some approaches to the problem of how best to sample and estimate from these flow records.

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