This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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. (Timehomogeneity) 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...
View
Full
Document
This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell.
 Summer '08
 WEBER

Click to edit the document details