1
Poisson Processes
An
arrival
is simply an occurance of some event — like a phone call, a job o
ff
er,
or whatever — that happens at a particular point in time.
We want to talk
about a class of continuous time stochastic processes called
arrival processes
that describe when and how these events occur.
Let
Ω
be a sample space and
P
a probability.
For any
ω
∈
Ω
, for all
t
we de
fi
ne
N
t
(
ω
)
as the number of arrivals in the time interval
[0
, t
]
given the
realization
ω
.
We call
N
=
{
N
t
, t
≥
0
}
an arrivial process.
Clearly, as time
evolves
N
t
(
ω
)
jumps up by integer amounts with new arrivals.
The type of
arrivial process we are most interested in is called a
Poisson process
. A Poisson
process is de
fi
ned as an arrival process that satis
fi
es the following three axioms:
1. for almost all
ω
, each jump is of size 1;
2. for all
t, s >
0
,
N
t
+
s
−
N
t
is independent of the history up to
t
,
{
N
u
, u
≤
t
}
;
3. for all
t, s >
0
,
N
t
+
s
−
N
t
is independent of
t
.
The
fi
rst axiom says that there is a zero probability of two arrivals at the
exact same instant in time; the second says that the number of arrivals in the
future is independant of what happened in the past; and the third says the
number of arrivals in the future is indentically distributed over time, or that
the process is stationary.
What is interesting is that these simple qualitative
feeatures of the process imply the following:
Lemma 1
If
N
is a Poisson process then for all
t
≥
0
,
P
(
N
t
= 0) =
e
−
λt
for
some
λ
≥
0
.
Proof.
By the independence axiom,
P
(
N
t
+
s
= 0) =
P
(
N
t
= 0)
P
(
N
t
+
s
−
N
t
= 0)
. By the stationarity axiom,
P
(
N
t
+
s
−
N
t
= 0) =
P
(
N
s
= 0)
. Hence, if
we let
f
(
t
) =
P
(
N
t
= 0)
, we have just established
f
(
t
+
s
) =
f
(
t
)
f
(
s
)
.
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 Spring '02
 Krueger
 Probability theory, Exponential distribution, Poisson process, Tn, CDF

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