Lecture_18

# Lecture_18 - 1 POISSON PROCESS Lecture 18 ORIE3500/5500...

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Unformatted text preview: 1 POISSON PROCESS Lecture 18 ORIE3500/5500 Summer2009 Chen Class Today • Poisson Process • Geometric and Negative Binomial Distribution 1 Poisson Process The Poisson distribution is used heavily to model queues, number of phone call received in a time period etc. The Poisson process is used in queuing theory a lot. It has the power to model queue over time. A Poisson process { N ( t ) : t ≥ } is a stochastic process and hence has random variables indexed by time, which satisfies 1. (Independence) The number of events occurring in two disjoint inter- vals, are independent random variables. 2. (Time-homogeneity) For every t and h , the number of events occurring between t and t + h depends only on h and has distribution P [ N ( t + h )- N ( t ) = k ] = e- λh ( λh ) k k ! ,k = 0 , 1 , 2 ,.... which means that the number of events occurring in any interval with length h , denoted by N h , does not depend on the starting point t and N h ∼ Poi ( λh ) . Note that a Taylor series expansion of e- λh yields e- λh = 1- λh + o ( h ) , here, o ( h ) means that lim h → o ( h ) h = 0 . Therefore, we have p N h (0) = e- λh = 1- λh + o ( h ) and p N h (1) = λhe- λh = λh- ( λh ) 2 + o ( h ) = λh + o ( h ) . For the other k = 2 , 3 ,..., we just have p N h ( k ) = o ( h ) . This tells us that for an interval with length h small enough 1 1 POISSON PROCESS • The probability of a single event occurring is roughly λh , plus a negli- gible term; • The probability of no event occurring is roughly 1...
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## This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell.

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Lecture_18 - 1 POISSON PROCESS Lecture 18 ORIE3500/5500...

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