随机过程--Chapter 2

customer in (0,t] is t-Si. Adding the revenues generated by all
arrivals in (0,t]
N (t)
N (t )
(t Si ) ,
i 1
E (t Si )
i1
14
2.2 Properties of Poisson processes
Solution:
(a) E[S10]=10/= 10 days
(b) P{X11>2} = e -2 = e-2 0.1333
9
2.2 Properties of Poisson processes
Arrival time distribution
Proposition 2.2 :
The arrival time of the nth event Sn follows a Γ distribution
f (t) e t
each interarrival time {Xn, n1} follows an exponential
distribution with parameter .
8
2.2 Properties of Poisson processes
Example 1 Suppose that people immigrate into a territory at a Poisson
and X1 and X2 are independent
7
2.2 Properties of Poisson processes
Similarly, we obtain: P{ Xn>tXn-1=s} = e-t P{ Xn tXn-1= s} = 1- e-t
We obtain the interarrival time distribution:
n
E i1
Si
N (t )
n
n E U(i)
The counting process {N(t), t0} is said to be a Poisson process
having arrival rate , >0, if
(a) N(0) = 0 (b) The process has independent increments (c) The number of events in any interval of length t is
yi
)
n! tn
,
0 y1 ... yn t
11
2.2 Properties of Poisson processes
Proposition 2.3:
Given that N(t)=n, the n arrival times S1… Sn have the same distribution as the order statistics corresponding to the n i.i.d. samples from U(0,t). that is,
Let Sn denotes the epoch of the nth arrival of N and
define S0=0. The interval time Xn= Sn-Sn-1, so
n
Sn X k ,
n 1,2......
k 1
Proposition(命题） 2.1:
A Poisson process N={N(t), t0} with rate , the interarrival
time {Xn, n1}are independently and identically distributexponential distribution
with parameter .
( f(t)=e-t , t>0, mean=1/ )
Example 2: (2.2.3)
A cable TV company collects $1/unit time from each subscriber. Subscribers sign up in accordance with a Poisson process with
rate . What is the expected total revenue received in (0,t]?
12
2.2 Properties of Poisson processes
=
h e ... h e e h1 1
hn1 (t h1 ...hn ) n
et (t)n
(According to Eq.2-1-1)
n!
n!
= t n h1...hn
P{ti Si t+hi, i=1,…nN(t) = n} =
(a) N(0) = 0, (b) It satisfies the stationary and independent increment
properties
(c) P{N(h)=1}=h+o(h) (d) P{N(h) 2}=o(h)
3
2.1 Definition
Definition 2:
rate =1 per day.
(a) What is the expected time until the tenth immigrant arrives? (b) What is the probability that the elapsed time between
the tenth and the eleventh arrival exceeds two days?
find the previous expectation by conditioning on N(t)
N(t)
E (t
Si
)
N
(t)
n
E
n
(t
Si
)
N (t )
n
nt
E
n
Si
N(t) n
i1
i1
i1
Let U1,…Un be iid random variables which follow U(0,t). so
n! tn
h1
.
.
.hn
P{ti
Si
ti
hi , i 1...n h1....hn
N (t )
n}
n! tn
taking the limits as hi 0 for all i, we obtain
n!
f S1.....Sn N (t)(t1 ,...,tn n) t n
13
2.2 Properties of Poisson processes
f (t) e t X1 follows an exponential distribution with parameter
6
2.2 Properties of Poisson processes
For any s>0 and t>0,
{X2>tX1=s} {0 event in (s, s+t]X1=s} P{X2>tX1=s} = P{0 event in (s, s+t]X1=s}
= P{0 event in (s, s+t]} (independent-increment) = P{0 event in (0, t]} (stationary-increment) = P{N(t)=0}= e-t P{ X2 tX1= s} = 1- e-t
f (t) e t
X2 follows an exponential distribution with parameter ,
f (t) e t (t) n1
(n 1)!
10
2.2 Properties of Poisson processes
Past arrival times given
– Joint density of past arrival times
Order statistics Let Y1, Y2……Yn are n random variables, if we arrange these random variables from small to big, note Y(1) = y1 is the smallest in the sequence, Y(2) = y2 is the second smallest, …. Y(n) = yn is the biggest in the sequence. Y(1) < Y(2)……< Y(n) , Y(1)……Y(n) or y1……yn are the order statistics of Y1…Yn.
Solution:
(Depends on the total number of subscribers and their arriving time)
Let N(t) denote the number of subscribers, and Si denote the arrival time of the ith customer. The revenue generated by this
Let f is density of distribution of Yi, if f follows the uniform
density over (0,t), the joint density of {Y(i)} is :
n
f
Y()1
.....Y(
n
(
)
y1
,...,
yn )
n!
i 1
f
(
5
2.2 Properties of Poisson processes
Proof: {X1>t} {N(t)=0} P{X1>t} = P{N(t)=0} = e-t
P{ X1 t} = 1- e-t differentiating the two sides of equation with respect to t, We obtain the density of distribution of X1:
Poisson distributed with mean t. t0
P{N (t) n} et (t)n ,
n!
n 0,1,2,...... (Eq.2-1-1)
E[N(t)] = t
4
2.2 Properties of Poisson processes
Interarrival time distribution
Stochastic Processes
Ma Shixiang HIT Shenzhen Graduate School
1
Chapter 2 Poisson processes
2.1 Definition 2.2 Properties of Poisson processes 2.3 Nonhomogeneous Poisson processes 2.4 Compound Poisson processes 2.5 Filtered Poisson processes
with parameter (n,).
( f (t) e t (t)n1 )
(n 1)!
Proof:
{Sn t} {N(t)n}
P{Sn t}= P{N(t)n} =
kn
e t (t)k
k!
differentiating the two sides of equation with respect to t :
n! f S1.....Sn N (t)(t1,...,tn n) t n , 0 t1 ... tn t
Proof:
P{ti Si t+hi, i=1,…nN(t) = n}
=
P{one event in (ti ,ti
hi ],
1 i n, no events elsewhere in (0, t)} P{N (t) n}
2
2.1 Definition
Consider a counting process N={N(t), t0}, where N(t) denotes the number of arrivals in the interval (0,t].
Definition 1
A counting process N ={N(t), t 0} is a Poisson process with rate >0, if it possesses the following properties: