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

IE502: Probabilistic Models IEOR @ IITBombay

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 λt, i.e. for all s, t ≥ 0,

• To show that arbitrary counting process is a Poisson

process, we must show that conditions (i), (ii) and (iii) are satisfied

( ) ( )

{ }

( ) ,

0,1,... !

n t

e t P N t s N s n n n

λ

λ

−

+ − = = =

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
• The function f(.) is said to be o(h) if

) ( lim =

→

h h f

h

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?