Poisson Process Guide: Properties, Distributions and Examples

Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PP

AMA528
PROBABILITY AND STOCHASTIC MODELS
DEPARTMENT OF APPLIED MATHEMATICS

Lecturer & Tutor: Dr. Catherine LIU
Contact: 2766 6931 (O); Ofﬁce Venue: HJ616
Consultation Hours: 7:45pm-8:45pm, Mon. & 4:00pm-5:00pm, Tues.

16/11/2011

AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 1 / 15


Chapter 7

The Poisson Process



Outline

1
Review of exponential Distribution
2
Counting Process
3
Poisson Process
4
Nonhomogeneous Poisson Process
5
Compound Poisson Process



Review on exponential distribution
pdf: 0-rate parameter, Y exp ; 0.
y 1 x
e if y 0 e if x 0
f y g x
0 elsewhere 0 elsewhere

2 2
E Y 1 Var Y 1 ;E X Var X .
cdf:
y x
1 e if y 0 1 e if x 0
F y G x
0 elsewhere 0 elsewhere

t 1 1
MGF: MY t 1 ; MX s 1 s .
Memoryless: For all s t 0,

P X s tX t P X s or P X s t P X s P X t

.



A note

Suppose X1 Xn i.i. and Xi exp i for i 1 n.
n
P min X1 Xn x P X1 x exp i x
i 1

i
P Xi Xj
i j

Eg1 (example 5.5, pp.2): Suppose one has a stereo system consisting of two main
parts, a radio and a speaker. If the lifetime of the radio is exponential with mean 1000
hours and the lifetime of the speaker is exponential with mean 500 hours independent
of the radio’s lifetime, then what is the probability that the system’s failure (when it
occurs) will be caused by the radio failing?



Counting Process

A stochastic process N t t 0 is said to be a counting process if N t represents
the total number of ”events” that have occurred up to time t if
1 N t 0;
2 N t is integer valued;
3 If s t, then N s N t ;
4 For s t, N t N s equals the number of events that have occurred in the
interval s t .
Eg: Let N t equal # of persons who enter 7-11 shop at or prior to (or by) time t, then
N t t 0 is a counting process;
But if N t equal # of persons in the store at time t, then N t t 0 would not be a
counting process.
Independent increments: if # of events that occur in disjoint time intervals are
independent.
Eg: N 10 N 3 is independent of N 15 N 10 .



Poisson Process

A counting process N t t 0 is said to be a Poisson Process having rate 0,
if
1 N 0 0;
2 The process has independent increments;
3 The # of events in any interval of length t is Poisson distributed with mean t.
That is, for all s t 0,
n
t t
P N t s N s n e n 0 1
n



Alternative deﬁnition

f x f x
Inﬁnite small o 1 : f x o x or x
o 1 or lim x
0.
x 0

A counting process N t t 0 is said to be a Poisson Process having rate 0,
if
1 N 0 0;
2 The process has independent and stationary increments;
3 P N h 1 h o h ;
4 P N h 2 o h .



Interarrival and waiting times
Consider a Poisson process: Let T1 be the time of the ﬁrst event; Let Tn denote the
elapsed time between the n 1 st and the nth event for n 1.
The sequence Tn n 1 2 is called the sequence of interarrival times.
Distribution of Tn : Tn n 1 2 i.i.d. exp .
n
Let Sn Ti , n 1, the arrival time of the n-th event, then Sn is called
i 1
the waiting time until the n-th event.
Distribution of Sn : Sn Gammar n . The pdf of Sn is
n 1
t t
fsn t e I t 0
n 1

A useful result:
Sn t N t n
Remark: Based on Tn n 1 with rate , we can set up a Poisson process with rate .



Property 1

Suppose two Poisson processes
1
N1 t t 0 with rate 1 ; Sn : the time of the n-th event of the 1st process;
2
N2 t t 0 with rate 2; Sm : the time of the m-th event of the 2nd process.
N1 t t 0 and N2 t t 0 are independent.
n m 1 k n m 1 k
1 2 n m 1 1 2
P Sn Sm
k 1 2 1 2
k n



Property 2

Let X1 Xn i.i.d. U 0 t and the corresponding order statistics X 1 Xn.
Then S1 Sn N t n X1 Xn .
That is, the conditional joint pdf of S1 Sn given that N t n is

n
f s1 sn n 0 s1 sn t
tn



Nonhomogeneous Poisson Process

Nonstationary Poisson proess: allow the arrival rate at time t to be a function of t.
A counting process N t t 0 is said to be a nonhomogeneous Poisson Process
with intensity function t t 0, if
1 N 0 0;
2 The process has independent increments;
3 P N t h N t 1 t h o h ;
4 P N t h N t 2 o h .
t
Let m t 0
y dy. Then m t is called the mean value function of the
nonhomogeneous Poisson process. And
n
m s t m s m s t m s
P N s t N s n e n 0
n



Example 2

Eg 2 (example 5.20, pp.15): Siegbert runs a hot dog stand that opens at 8 am. From 8
am until 11am customers seem to arrive, on the average, at a steadily increasing rate
that starts with an initial rate of 5 customers per hour at 8 am and reaches a maximum
of 20 customers per hour at 11 am. From 11 am until 1 pm the (average) rate seems
to remain constant at 20 customers per hour. However, the (average) rate seems to
remain constant at 20 customers per hour. However, the (average) arrival rate then
drops steadily from 1pm until closing time at 5pm at which time it has the value of 12
customers per hour. If we assume that the numbers of customers arriving at
Siegbert’s stand during disjoint time periods are independent, then what is a good
probability model for the above? What is the probability that no customers arrive
between 8:30 am and 9:30 am on Monday morning? What is the expected number of
arrivals in this period?



Compound Poisson Process
Let Yi i 1 be a family of i.i.d. r.v.s which are independent of a Poisson process
N t t 0 .
N t
Let X t Yi t 0. Then the r.v. X t is said to be a compound Poisson r.v. and
i 1
the stochastic process X t t 0 is said to be a compound Poisson process.
Eg: Let N(t) be # of customers leave a supermarket by time t distributed with
Poisson t . Let Yi i 1 2 , the amount spent by the i-th customer, i.i.d. Let X t
N t
be the total amount of money spent by time t. Then X t Yi t 0. And
i 1
X t t 0 is a compound Poisson process.
Remark: Let Yi 1, the X t N t , a usual Poisson process.

X t X t N t N t Y1 t Y1
2
Var X t Var X t N t Var X t N t t Y1



Example 3

Eg 3 (exampel 5.22, pp. 19) Suppose that families migrate to an area at a Poisson

rate 2 per week. if the number of people in each family is independent and takes

on values 1, 2, 3, 4 with respective probabilities 1 , 3 , 1 , 6 , then what is the expected
6
1
3
1

value and variance of the number of individuals migrating to this area during a ﬁxd

ﬁve-week period?


Poisson Process Guide: Properties, Distributions and Examples

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (10)

Similar to Poisson Process Guide: Properties, Distributions and Examples

Similar to Poisson Process Guide: Properties, Distributions and Examples (11)

Recently uploaded

Recently uploaded (20)

Poisson Process Guide: Properties, Distributions and Examples