טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentIorsh Maxim
SubjectLarge Deviations for a Polling System with Exhaustive
Service
DepartmentDepartment of Applied Mathematics
Supervisor Professor Adam Shwartz


Abstract

This research thesis is dedicated to the analysis of a polling system with exhaustive service discipline, by means of the theory of Large Deviations.


We consider a model which expresses the polling system as a jump Markov process. While free Markov processes on Euclidean spaces have been extensively studied, this model presents a new challenge as it demonstrates an abrupt change in behavior every time one of the served queues becomes exhausted .


In order to deal with such discontinuities, we introduce a new topological space as the state space for the random process of clients arrival and service, where discontinuities in the behavior of the process are located on the boundaries. We apply the results which exist for the free processes, while taking special care of boundary areas, to deduce similar results for the considered model .


In the course of the research we establish the notion of a rate function, as a composition of rate functions for the free processes. An interesting outcome of our discussion is that sometimes the rate function doesn't have the usual structure of an integral of a local rate function over time, but rather exhibits some "predefined strategy" of choice between free rate functions . Our main achievement is the establishment of the Large Deviations Principle for the discussed model.


The achieved results provide a base for further research of polling systems with exhaustive service. They allow us to address the questions about the probability of escape from the stable state, the escape path, and more .