12 INTRODUCTION TO TRAFFIC MODELS AND ANALYSIS

1.2.1 NON-HOMOGENEOUS POISSON PROCESS

In many applications, the assumption that the point process is stationary might not be valid, and we

are forced to do away with the assumption that the increments are stationary. Of course, in order to

develop useful insights and to perform calculations, we need to impose some hypotheses. A natural

extension to the Poisson process is the non-stationary or non-homogeneous Poisson process deﬁned

by the following properties:

Deﬁnition 1.13 {N

t

}

t≥0

is said to be a non-homogeneous Poisson process with intensity λ

t

if:

1. N

0

= 0

2. For t>s, P(N

t

− N

s

= k) =

1

k!

t

s

λ

u

du

k

e

−

t

s

λ

u

du

3. {N

t+s

− N

s

} is independent of N

u

,u≤ s.

Of course, since λ

t

depends on t, we need to impose some conditions on the point process for

it to be well-behaved, i.e., the points do not accumulate. For this, it is necessary and sufﬁcient that

∞

0

λ

s

ds =∞.

With this property, it immediately follows that {lim

t→∞

N

t

=∞ a.s.}⇐⇒

lim

t→∞

t

0

λ

s

ds =∞and 0 <T

1

<T

2

< ···<T

n

<T

n+1

< ··· and once again:

P(N

t+dt

− N

t

= 1) = λ

t

dt + o(dt)

P(N

t+dt

− N

t

≥ 2) = o(dt)

Note the last property states that there cannot be more than one jump taking place in an

inﬁnitesimally small interval of time, or two or more arrivals cannot take place simultaneously.

We now see an important formula called Campbell’s Formula, which is a smoothing formula.

Proposition 1.14 Let {N

t

}

t≥0

be a non-homogeneous Poisson process with intensity λ

t

. Let {X

t

} be

a continuous-time stochastic process such that N

t

− N

s

is independent of X

u

,u≤ s and let f (.) be a

continuous and bounded function. Then:

E[

N

t

n=1

f(X

T

n−

)]=E[

t

0

f(X

s

)λ

s

ds] (1.6)

Proof. It is instructive to see the proof. Let p

n

(t) be the probability density of the n-th point T

n

(which is given by the density of the Erlang-n distribution).Then, from (1.5) noting that the Poisson

process has the intensity λ

t

,

p

n

(t) = λ

t

(

t

0

λ

s

ds)

n−1

(n − 1)!

e

−

t

0

λ

s

ds

1.2. TRAFFIC ARRIVAL MODELS 13

from which it readily follows that

∞

n=0

p

n

(t) = λ

t

.

Therefore,

E[

N

t

n=1

f(X

T

n

−

)]=E[

∞

n=1

f(X

T

n

−

)1I

[T

n

≤t]

]

= E[

t

0

∞

n=1

f(X

s−

)p

n

(s)ds]=

t

0

E[f(X

s

)λ

s

]ds

where X

s−

can be replaced by X

s

because of continuity (in s).

Also since λ

t

is non-random it can be removed from the expectation. 2

Remark 1.15

1. Note

N

t

n=0

f(X

T

n

) can be written as

t

0

f(X

s

)N(ds) where the integral is the so-called

Stieltjes integral.

2. In particular, if N

t

is homogeneous Poisson with intensity, λ and X

t

is stationary then:

E[

t

0

f(X

s

)N(ds)]=λ

t

0

E[f(X

s

)]ds = λt E[f(X

0

)]

This is the so-called PASTA property that will be discussed a bit later.

3. More generally, a point process N is said to be a Poisson process in R

d

(d-dimensional)

with intensity λ(x), x ∈ R

d

if for any A, B ⊂ R

d

and A

B = φ then N(A) and N(B) are

independent, and the distribution of N(A) is given by the Poisson distribution with parameter

A

λ(x)dx.Let0 denote the origin. Suppose N is a Poisson process in

2

with intensity λ,

then the probability that there is no point within a radius r of 0 is just e

−πr

2

λ

i.e., the distance

to the nearest point is exponentially distributed.

Besides the obvious time dependence, non-homogeneous Poisson processes have another

important application in the modeling of networks. Consider the case of a so-called marked Poisson

process, which is the basic model that is used in the modeling of queues. This is characterized by

arrivals which take place as a Poisson process with rate or intensity say λ. Each arrival brings a mark

associated with it denoted by σ

n

which are assumed to be i.i.d. In applications, the mark denotes the

amount of work or packet length associated with an arriving packet. Then the process:

X

t

=

n

σ

n

1I

(0<T

n

≤t]

(1.7)

is called a Marked Poisson Process if the {T

n

} correspond to the points of a Poisson process.

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