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ECE & CSE
ECECSE 861:
Introduction to Computer
Communication Networks
Ness B. Shroff
ECE & CSE
Lectures 69
ECE & CSE
Arrival Process or Counting Process
Definition:
An
arrival process
or
counting process
is that
stochastic process {N(t); t
≥
0} such that:
1)
N(0) = 0
2)
N(t) is integer valued
3)
N(t) is nondecreasing; i.e., if s
≤
t, then N(s)
≤
N(t).
4)
For s<t, N(t +s) – N(t) equals the number of events (or arrivals) in
(s,t].
(right continuous property)
Time
N(t)
t
1
t
2
t
3
ECE & CSE
Poisson Process
Definition I:
The arrival process {N(t), t
≥
0} is said to be a Poisson Process with rate
λ
>0 , if
1)
N(0) = 0
2)
P{N(t) = 1 } =
λ
t + o(t)
3)
P{N(t)
≥
2} = o(t)
4)
For any t
≥
0, s
≥
0,
1)
N(t +s) – N(t) is independent of {N(u): u
≤
t}.
2)
This is known as the
Independent Increments property.
5)
For any t , s
≥
0, the distribution of N(t+s) – N(t) is independent of t.
1)
This is known as the
Stationary Increments property.
ECE & CSE
Poisson Process (Cont’d)
Remember: A function f is said to be o(h) if
Equivalent Definition II:
The arrival process {N(t), t
≥
0 } is said to be Poisson Process
with rate
λ
, if
1)
N(0) = 0
2)
Independent Increments
3)
The number of arrivals in any interval of length s is Poisson
distributed with mean
λ
s,
for all s, t
≥
0.
0
)
(
lim
0
=
→
h
h
f
h
!
)
(
n}
N(t)
–
s)
P{N(t
n
s
e
n
s
λ
λ
−
=
=
+
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ECE & CSE
Independent Increments Property
N(t + s) – N(t) is independent of {N(u): u
≤
t}. Arrivals
in the future (beyond time t) are independent of the
entire past history up to time t (
very strong property
).
t
ECE & CSE
Stationary Increments Property
Strong Property (but less so than Ind. Inc):
For any t, s
≥
0 the distribution of N(t+s) – N(t) is independent
of t.
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 Spring '11
 shroff
 Poisson Distribution, Exponential distribution, Poisson process, Lévy process, Counting process

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