Download Electrical Noise

X1i
0.3
X2i
Applications of Poisson Process
X3i
0.2
X4i
X5i
0.1
Wang C. Ng
0
0
2
4
6
8
10
12
14
16
10
12
14
16
i
1
.4
X1i
y2i
X2i
X3i
0.6
y3i
0.2
0.4
y4i
X4i
X5i
y5i
0.1
0
0
0.2
0
0
1
y1i
0.8
y1i
0.3
0.8
1
2
4
6
8
i
10
12
14
0
0
16 0
16 0
2
4
6
8
i
16
Telephone traffic
• Pure chance traffic: Independent random
events (memoryless).
• Stationary: Busy/peak hours only.
• The number of calls follows the Poisson
distribution.
P( x) 
x
x!
e   , where  is the mean number of calls.
Example:
• On average, one call arrives every 5
seconds. During a period of 10 seconds,
what is the probability that:
–
–
–
–
no call arrives?
one call arrives?
two calls arrive?
more than two calls arrive?
Solution
P( x) 
P(0) 
P(1) 
P(2) 
x
x!

0
0!

1
1!

e   , where   2
e  2  e  2  0.135
e  2  2e  2  0.270
2
e
2
 2e
2
 0.270
2!
P( 2)  1  P(0)  P(1)  P(2)
 1  0.135  0.270  0.270  0.325
Telephone traffic
• The interval between calls follows the
exponential distribution.
P(T  t )  e t / T ,
where T is the mean time between calls.
• The call duration also follows the
exponential distribution.
t / h
P(T  t )  e ,
where h is the mean call duration.
Example:
• Average call duration is 2 minutes. A call
has already lasted 4 minutes. What is the
probability that:
– the call last at least 4 more minutes?
– the call will end within the next 4 minutes?
t / h
2
P(T  t )  e
 e  0.135
P(T  t )  1  P(T  t )
 1 e
2
 1  0.135  0.865
Telephone traffic
• The number of calls in progress, assuming
infinite (large) number of trunks (circuits)
carrying the call, also has a poisson
distribution.
Ax  A
P( x) 
e ,
x!
where A is the mean number of calls arriving
during the average holding time (duration) .
Example:
• Average call duration is 2 minutes and the
mean number of calls per minute is 3. What
is the probability that
– 2 calls are in progress?
– More than 2 calls are in progress?
Solution
A  3 2  6
2
6 6
P(2)  e  0.045
2!
P( 2)  1  P(0)  P(1)  P(2)
0
1
2
6 6 6 6 6 6
 1 e  e  e
0!
1!
2!
6
6
6
 1  e  6e  18e
 1  (1  6  18)e
6
 0.938
Poisson modeling
• Poisson model has been used to study
network traffic
• It has attractive theoretical properties
• It has been studied thoroughly
• It has represented the telephone traffic well
The failure of the Poisson model
• However, recent studies have shown that
the Poisson model is inadequate for many
types of internet traffic (see attached article)
• In general, the Poisson model fails to
represent the “bursty” nature of internet
traffic
• A new model has been proposed to replace
the Poisson model
Self-similar process
• In the internet traffic studies the properties
of self-similarity has been observed.
• This type of processes can be analyzed
using the recently developed chaos and
fractal theory