Download 3 poisson processes

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3.1
POISSON PROCESSES
EXPONENTIAL DISTRIBUTION
First remind some facts on exponential distribution.
Definition 1. A non-negative random variable X is said to be an exponential
random variable with parameter λ (or mean 1/λ), denoted by X ∼ exp(λ), if its
cumulative distribution function F (x) satisfies
F (x) = P (X ≤ x) =
¿ 0
if x ≤ 0,
1 − e−λx if x > 0.
(3.1)
The most important property of the exponential distribution is the memoryless property
P (X > t + s|X > s) = P (X > t).
(3.2)
For instance, if X represents the lifetime of a certain computer component, then Equation
(3.2) expresses the fact that the probability that an s year old component will last another
t years is the same as the probability that a new component will last t years. It is as if
the component has no memory taht it has already been functioning for s years – it does
not show any signs of aging.
The memoryless property (3.2) can be derived in the following way:
P (X > t + s|X > s) =
=
P (X > t + s, X > s)
P (X > t + s)
=
=
P (X > s)
P (X > s)
e−λ(t+s)
= e−λt = P (X > t).
e−λs
Moreover, the following theorem holds.
Theorem 2. The only continuous random variable with the memoryless property is
the exponential random variable.
More on exponential distribution is available at
➥ http://www.dartmouth.edu/~chance/teaching_aids/books_articles/
probability_book/Chapter5.pdf
Lecture notes for the course Stochastic Models at FD ČVUT.
Prepared by Magdalena Hykšová, Prague, 2003.
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3. POISSON PROCESSES
3.2
POISSON PROCESS
A Poisson process is frequently used as a model for counting events occurring one at a
time.
Definition 2. Let {Xn , n ≥ 1} be a sequence of random variables representing the
inter-event times. Define
S0 = 0
Sn = X1 + X2 + · · · + Xn .
(3.3)
Then Sn is the time of occurrence of the n-th event. Define
N (t) = max{n ≥ 0 : Sn ≤ t},
t ≥ 0,
(3.4)
to be the number of events that have taken place during (0, ti. Then
{N (t) : t ≥ 0}
(3.5)
is called a counting process.
Definition 3. If the {Xn , n ≥ 1} is a sequence of independent and identically
distributed random variables under an exponential distribution exp(λ), the counting
process {N (t) : t ≥ 0} defined by Equation (3.4) is called a Poisson process with
parameter λ and is denoted by P P (λ).
Remind that random variables X1 , X2 , · · · , Xn are called independent and identically distributed random variables if they are
• mutually independent, that is
Fn (x1 , x2 , . . . , xn ) = FX1 (x1 )FX2 (x2 ) · · · FXn (xn ),
(3.6)
where
Fn (x1 , x2 , . . . , xn ) = P (X1 < x1 , X2 < x2 , . . . , Xn < xn )
is a joint cumulative distribution function of (X1 , X2 , · · · , Xn ) and FXi (·) is a probability density function of Xi , and
• for all x it is
FX1 (x1 ) = FX2 (x2 ) = · · · = FXn (xn ) .
(3.7)
Let {N (t) : t ≥ 0} be a real-valued stochastic process. For a fixed s ≥ 0 and t ≥ 0,
X(t + s) − Xs
is called the increment over the interval (s, s + t).
3.2. POISSON PROCESS
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Definition 4. A stochastic process {N (t) : t ≥ 0} is said to have stationary and
independent increments if
(i) the distribution of X(t + s) − Xs is independent of s,
(ii) the increments over nonoverlapping intervals are independent.
The following theorem provides the the necessary and sufficient conditions for a stochastic process to be a Poisson process; these conditions can be taken as an alternate
definition of a Poisson process.
Theorem 3. A stochastic process {N (t) : t ≥ 0} is a Poisson process with parameter
λ if and only if
(i) it has stationary and independent increments,
(ii) P (N (t) = k) = e−λt
(λt)k
for t ≥ 0, k = 0, 1, 2, . . .
k!
A detailed exposition of Poisson processes including practical examples can be found
in the following Interned addresses:
➥ http://keskus.hut.fi/opetus/s38143/2001/luennot/E_lect06.pdf
➥ http://asrl.ecn.uiowa.edu/dbricker/Stacks_pdf1/Poisson%
20processes%20A.pdf
➥ http://asrl.ecn.uiowa.edu/dbricker/Stacks_pdf1/Poisson_Ppties.pdf
➥ http://www.columbia.edu/~ks20/4106-03-summer/4106-03-Notes-6.pdf