ISEN 609 Lecture 8

ISEN 609 Lecture 8 - 8. Poisson Process: first some...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
8. Poisson Process: first some background Poisson distribution: Where does this formula come from? Poisson distribution is an “approximation” to the Binomial distribution. Y Bin [ n, p ] = P ( Y = k ) = ± n k ² p k (1 - p ) n - k , k = 0 , 1 , . . . , n X Pois [ λ ] = P ( X = k ) = e - λ λ k k ! , k = 0 , 1 , . . .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Poisson Approximation to Binomial Let’s look at what happens to Bin [ n , p ] as n gets large and p get small (such that the product np is held constant - call it ): λ n → ∞ : n ( n - 1) · · · ( n - k + 1) n k 1 , ± 1 - λ n ² n e - λ , ± 1 - λ n ² k 0 P ( Y = k ) λ k e - λ k ! , k = 0 , 1 , . . . P ( Y = k ) = ± n k ² p k (1 - p ) n - k = n ! k !( n - k )! ³ λ n ´ k ³ 1 - λ n ´ n - k = n ( n - 1) · · · ( n - k + 1) n k λ k k ! (1 - λ n ) n (1 - λ n ) k
Background image of page 2
“Memoryless” property We say that a random variable has the memoryless property if: What does this property say? Suppose the random variable Z represents the lifetime of an engineered product P ( Z > t + s | Z > s )= P ( Z > t )
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Exponential Distribution Suppose . Then for In fact, the exponential is the only continuous r.v. with this property. P ( Z > t + s | Z > s ) = P ( Z > t + s, Z > s ) P ( Z > s ) = P ( Z > t + s ) P ( Z > s ) = e - μ ( t + s ) e - μs = e - μt = P ( Z > t ) Z exp ( μ ) t, s > 0
Background image of page 4
Exponential Distribution Let X and Y be independent random variables with parameters , respectively. Find the distribution of min( X,Y ) max( X,Y ) X + Y Find P(X > Y) μ and ν
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Counting Processes <--> Point Process Many of the models we develop involving characterizing occurrences that happen over time. Suppose we are counting the number of occurrences of a particular event over time, starting at a point t = 0. Let N ( t ) be the number of events we have observed up to time t ; i.e., the number of events in [0, t ]. is called a counting process . Counting processes make up some of the most common models in engineering - arrivals to a service system, flaws in a manufacturing process, bugs in a piece of software, etc. { N ( t ) , t 0 }
Background image of page 6
And Now: Introducing The Poisson Process If random occurrences happen over time according to certain assumptions - namely, stationarity, independent increments, and orderliness - and if we start counting at time 0, and if 0 < P ( N ( t ) = 0) < 1, then there exists a positive number such that Note that and therefore and λ P ( N ( t ) = n ) = ( λ t ) n e - λ t n ! , n = 0 , 1 , 2 , . . . , t > 0 N ( t ) Pois [ λ t ] , E [ N ( t )] = λ t λ = E [ N (1)]
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Relationship between Counts, Interarrival Times, and Occurrence Times Note that if we know the counting process , then we can determine the sequence of interarrival times as well as the sequence of occurrence times (Ross calls these waiting times) ; these are related by = n th jump in the counting process So to describe a point process model, we need to make assumptions about a counts, or waiting times, or interarrival times. { N ( t ) , t 0 } { S i ,i =1 , 2 , . . . } { T i , 2 , . . . } S n S n = n ± i =1 T i
Background image of page 8
The Poisson Process A orderly , stationary counting process with independent increments is called a Poisson process.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/28/2011 for the course ISEN 609 taught by Professor Klutke during the Spring '08 term at Texas A&M.

Page1 / 33

ISEN 609 Lecture 8 - 8. Poisson Process: first some...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online