Poisson Process• Point process is defined by random points in time 𝑡!• Counting process 𝑁 𝑡 = the number of occurrences of some event

in the time period [0, 𝑡]

𝑁 𝑡 = (!"#

$

𝑢(𝑡 − 𝑡!)

𝑁 0 = 0Interarrival times 𝑇! = 𝑡! − 𝑡!%#

• Common assumptions:– Non-overlapping intervals lead to independent increments– Stationary increments – distribution independent of time origin

• If for small ℎ we assume:– P 𝑁 ℎ = 1 = 𝜆ℎ + 𝑜 ℎ– P 𝑁 ℎ > 1 = 𝑜 ℎ

where 𝑓 ℎ is 𝑜 ℎ if 𝑙𝑖𝑚!⟶#$(!)!= 0

– Grinding away yields a Poisson pmf dependent upon time

Prob 𝑁 𝑡 = 𝑛 ≡ 𝑃' 𝑡 = 𝑒()*𝜆𝑡 '

𝑛!More generally

𝑃' 𝑡+ − 𝑡, = 𝑒() *!(*"𝜆 𝑡+ − 𝑡,

'

𝑛!

• Expectations:– Mean and variance – since a Poission RV

E 𝑁 𝑡 = Var 𝑁 𝑡 = 𝜆𝑡Clearly not WSS

– Correlation function: assume wlog 𝑡+ ≥ 𝑡,

E 𝑁 𝑡, 𝑁 𝑡+ = E 𝑁 𝑡, 𝑁 𝑡, + 𝑁 𝑡+ −𝑁 𝑡,= E 𝑁+ 𝑡, + E 𝑁 𝑡, 𝑁 𝑡+ −𝑁 𝑡,

= E 𝑁+ 𝑡, + E 𝑁 𝑡, E 𝑁 𝑡+ −𝑁 𝑡,= 𝜆𝑡, + 𝜆𝑡, + + 𝜆𝑡,𝜆 𝑡+ − 𝑡,

= 𝜆𝑡, + 𝜆+𝑡,𝑡+

In general E 𝑁 𝑡, 𝑁 𝑡+ = 𝜆min 𝑡,, 𝑡+ + 𝜆+𝑡,𝑡+

• Consider the delay until the first arrival, 𝑇,

P 𝑇, > 𝑡 = 𝑃# 𝑡 = 𝑒()*

𝑓, 𝑡 =𝑑𝑑𝑡 P 𝑇, < 𝑡 =

𝑑𝑑𝑡 1 − P 𝑇, > 𝑡

=𝑑𝑑𝑡 1 − 𝑒

()* = 𝜆𝑒()*

– i.e. an exponential random variable– Same for all inter-arrival times

• Consider the conditional probability of waiting 𝑠 more:

P 𝑇, > 𝑡 + 𝑠|𝑇, > 𝑡 =𝑒() *-.

𝑒()*= 𝑒().

i.e. is “memoryless” in that the future wait’s distribution is not a function of the prior wait duration.

• Joint probability:

P 𝑁 𝑡, = 𝑘,, 𝑁 𝑡+ = 𝑘+

= P 𝑁 𝑡, = 𝑘,, 𝑁 𝑡+ −𝑁 𝑡, = 𝑘+ − 𝑘,

= P 𝑁 𝑡, = 𝑘, P 𝑁 𝑡+ −𝑁 𝑡, = 𝑘+ − 𝑘,

= 𝑒()*" )*" #"

/"!𝑒() *!(*" ) *!(*"

#!$#"

(/!(/")!

= 𝑒()*!𝜆/! 𝑡,/" 𝑡+ − 𝑡, /!(/"

𝑘,! (𝑘+ − 𝑘,)!

• Also, condition on exactly one event in [0, 𝑇]:

P 𝑇, < 𝑡|𝑁 𝑇 = 1 =P 𝑇, < 𝑡,𝑁 𝑇 = 1

P 𝑁 𝑇 = 1

=𝑃, 𝑡 𝑃# 𝑇 − 𝑡

𝑃, 𝑇

=𝜆𝑡𝑒()*𝑒() 1(*

𝜆𝑇𝑒()1=𝑡𝑇

i.e. is uniform ! This generalizes to: given 𝑘 arrivals in [0, 𝑇]they are independent uniform random variables. Useful for simulation

• Example of use:– Customers arrive at a train station starting at time 0 with

rate 𝜆– The train leaves at time 𝑇– Find the average sum of the waiting times of all of the

customers

𝐸 "!"#

$(&)

𝑇 − 𝜏! = 𝐸( 𝐸 "!"#

$(&)

𝑇 − 𝜏! |𝑁 𝑡 = 𝑛

= 𝐸( 𝑛𝑇 − 𝐸 "!"#

(

𝜏! = 𝐸( 𝑛𝑇 − 𝑛𝑇2= 𝐸( 𝑛

𝑇2=𝜆𝑇)

2

• Extensions:– Non-homogeneous arrival rate 𝜆(𝑡)

• Define

𝜇 𝑡 = 0*

+𝜆 𝑠 𝑑𝑠

• Then

P 𝑁 𝑡) − 𝑁 𝑡# = 𝑛 = 𝑒, - +! ,- +"𝜇 𝑡) − 𝜇 𝑡#

(

𝑛!

– Sum of independent Poisson processes is a Poisson process

𝑀 𝑡 = "!"#

.

𝑁! 𝑡

𝜆/ = "!"#

.

𝜆!

– Compound processes – basically sum of a random number of unity values (typically iid)

𝑋(𝑡) = "!"#

$ +

𝑈!

can show (find conditional characteristic function for 𝑁 𝑡 = 𝑛 as the char fn of 𝑈! raised to the 𝑛th power and then average over 𝑛)

Φ0 + 𝜔 = "("*

1

Φ2 𝜔 (𝑒,3+𝜆𝑡 (

𝑛!

= 𝑒,3+ "("*

1𝜆𝑡Φ2 𝜔

(

𝑛!= 𝑒,3+𝑒,3+4# 5 = 𝑒,3+[4# 5 ,#]

• Mean 𝜇0 = E 𝑋 𝑡 = 𝜆𝑡𝜇2• Var 𝑋 𝑡 = 𝜆𝑡E 𝑈)

– Filtered Poisson Process

𝑋(𝑡) = "!"*

$ +

ℎ(𝑡 − 𝑡! , 𝑈!)

Φ0 + 𝜔 = exp 𝜆0*

+𝐸2 𝑒859(+,:) − 1 𝑑𝑠

– Renewal Theory – iid interarrival times, but not exponential

13.4.1 The arrivals of new telephone calls at a telephone switching offce is a Poisson process N(t) with an arrival rate of λ = 4 calls per second. An experiment consists of monitoring the switching offce and recording N(t) over a 10-second interval.(a) What is PN(1)(0), the probability of no phone calls in the first second of observation?(b) What is PN(1)(4), the probability of exactly four calls arriving in the first second of observation?(c) What is PN(2)(2), the probability of exactly two calls arriving in the

first two seconds?

13.4.5 Customers arrive at the Veryfast Bank as a Poisson process of rate λ customers per minute. Each arriving customer is immediately served by a teller. After being served, each customer immediately leaves the bank. The time a customer spends with a teller is called the service time. If the service time of a customer is exactly two minutes, what is the PMF of the number of

customers N(t) in service at the bank at time t?

13.5.1 Customers arrive at a casino as a Poisson process of rate 100 customers per hour. Upon arriving, each customer must flip a coin, and only those customers who flip heads actually enter the casino. Let N(t) denote the process of customers entering the casino. Find the PMF of N, the number of

customers who arrive between 5 PM and 7 PM

13.5.3 A subway station carries both blue (B) line and red (R) line trains. Red line trains and blue line trains arrive as independent Poisson processes with rates λR = 0.15 and λB = 0.30 trains/min respectively. You arrive at the station at random time t and watch the trains for one hour.(a) What is the PMF of N, the number of trains that you count passing through the station?(b) Given that you see N = 30 trains, what is the conditional PMF of R, the number of red trains that you see?

13.5.4 Buses arrive at a bus stop as a Poisson process of rate λ = 1 bus/minute. After a very long time t, you show up at the bus stop.(a) Let X denote the interarrival time between two bus arrivals. What is the PDF fX(x)?(b) LetW equal the time you wait after time t until the next bus arrival. What is the PDF fW(w)?

13.7.6 N(t) is a Poisson process of rate λ = 1 and X0,X1,X2, . . . is an iidsequence of Gaussian (0, σ) random variables that are independent of N(t). Consider the process {Y(t)|t ≥ 0} defined by Y(t) = XN(t). Find the expected value µY(t) = E[Y(t)] and the covariance function CY(t, τ). (Assume |τ| < t.)