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- h ( h ) 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 ( h ) . Note that a Taylor series expansion of e- h yields e- h = 1- h + o ( h ) , here, o ( h ) means that lim h o ( h ) h = 0 . Therefore, we have p N h (0) = e- h = 1- h + o ( h ) and p N h (1) = he- h = h- ( h ) 2 + o ( h ) = h + 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 h , plus a negli- gible term; The probability of no event occurring is roughly 1...
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Lecture_18 - 1 POISSON PROCESS Lecture 18 ORIE3500/5500...

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