Poisson Process IE 502: Probabilistic Models Jayendran - PowerPoint PPT Presentation

Poisson Process IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR IE502: Probabilistic Models IEOR @ IITBombay

Poisson Process: Definition 1 A counting process { N ( t ), t ≥ 0} is a Poisson process with rate λ , λ > 0, if i. N (0) = 0 ii. The process has independent increments iii. The number of events in any interval of length t is Poisson distributed with mean λ t , i.e. for all s , t ≥ 0, ( ) , n − λ λ t e t { } ( ) ( ) + − = = = P N t s N s n n 0,1,... n ! • To show that arbitrary counting process is a Poisson process, we must show that conditions (i), (ii) and (iii) are satisfied IE502: Probabilistic Models IEOR @ IITBombay

But.. Why Poisson Process? • The Poisson process is the most widely used model for arrivals into a system. Reasons: – Markovian properties of the Poisson process make it analytically tractable – The Poisson process appears often in nature, when we are observing the aggregate effect of a large number of individuals or particles operating independently. – In many cases it is an excellent model. For example: Communication networks, gamma ray emissions etc. – In scheduling, resource allocation etc problems, the arrival process has a less affect on the final decisions → Hence Poisson process is a good approx., especially when we are interested in the mean performance. IE502: Probabilistic Models IEOR @ IITBombay

A definition • To show that arbitrary counting process is a Poisson process, we must show that conditions (i), (ii) and (iii) are satisfied • For that we need an equivalent definition of Poisson process • Before that… a definition f ( h ) = lim 0 • The function f (.) is said to be o ( h ) if h → h 0 IE502: Probabilistic Models IEOR @ IITBombay

Definition 1 and 2 are equivalent • Definition 1 implies Definition 2 • Definition 2 implies Definition 1 – Refer the book for a formal proof – We shall discuss an ‘looser argument’ • Revisit Definition 2 – Explicit assumption of ‘stationary increments’ can be eliminated by modifying (iii) and (iv). IE502: Probabilistic Models IEOR @ IITBombay

Poisson Process: Definition 2 - revisited A counting process { N ( t ), t ≥ 0} is a Poisson process with rate λ , λ > 0, if i. N (0) = 0 ii. The process has independent increments iii. P{ N ( t + h ) − N( t ) = 1} = λh + o ( h ) iv. P{ N ( t + h ) − N( t ) ≥ 2} = o ( h ) • In plain English, what does assumptions (iii) and (iv) mean? IE502: Probabilistic Models IEOR @ IITBombay