General counting processes

In this section counting processes, a special case of stochastic processes, are briefly introduced. The results reported below make intensive use of properties ofr.v.s and stochastic processes. The aims of this section are: to introduce some elements of counting-process theory, to set a general notation and to report properties that will be used further on. The proofs of the following statements can be found in the major books on stochastic processes (e.g. Kingman (1992), Chiang (1980)).

A stochastic process {X(t); t ∈ [0, ∞)} is a family of r.v.s describing an empirical process, whose development is governed by probability laws.

A counting process is a stochastic process where X(t) takes values in the natural set X(t) ∈ 0, 1, 2, . . . (Chiang,2007).

An arrival process is a sequence of increasing r.v.s 0 < S1 < S2 < . . . which represent the

times at which some repeating phenomenon occurs.

The Sn are called arrival epochs. We can also equivalently specify the arrival process by

specifying either the interarrival intervals

Xn= Sn− Sn−1 ∀n = 1, 2, . . .

(and therefore Sn=Pni=1Xi), or the counting process

{N (t); t > 0}

that represents the number of arrivals in the interval (0, t]. This process is related to the epochs by:

{Sn≤ t} = {N (t) ≥ n}

A particular case of an arrival process is the renewal process.

A renewal process is an arrival process for which the sequence of interarrival times X1, X2, . . .

A special counting process: the Poisson process

Definition 2. A homogeneous Poisson process is a renewal process in which the interarrival intervals have an Exponential distribution function; i.e., for some rate λ >0, each Xi has the

density fX(x) = λe−λx for x ≥0.

The main properties of this process are:

P1 homogeneous Poisson processes have stationary increments; P2 homogeneous Poisson processes have independent increments;

P3 the arrival epochs S1, S2, . . . for a homogeneous Poisson process are distributed according

to an Erlang distribution

fSn(t) =

λntn−1e−λt (n − 1)!

P4 ar.v. N(t), in the counting process {N (t); t > 0}, denoting the number of arrivals in (0, t], is a Poissonr.v. with probability mass function

f_{N (t)}(n) = (λt)

n_e−λt

n!

These properties follow from the definition of a homogeneous Poisson process. We report below two alternative definitions of the Poisson process.

Definition 3. A homogeneous Poisson counting process {N (t); t > 0} is a counting process that satisfies property P4 (i.e., has the Poisson probability mass function) and has the independent and stationary increment properties.

Following Definition 3, consider the number of arrivals in a very small interval (t, t + δt]. Given property P1, N (t, t + δt) has the same distribution as N (δt) and therefore:

P {N(t, t + δt) = 0} = e−λδ≈ 1 − λδ + o(δ) P {N(t, t + δt) = 1} = λδe−λδ ≈ λδ + o(δ) P {N(t, t + δt) ≥ 2} ≈ o(δ)

(2.3)

From which Definition 4 follows:

Definition 4. A homogeneous Poisson counting process is a counting process that satisfies Equation 2.3(i.e. the probability of having more than one event in a small interval approaches 0 as δ approaches 0) and has the stationary and independent increment properties

The Poisson process can be extended in a number of ways.

A thinned Poisson process can be constructed as follows. Let {N (t); t > 0} be a Poisson counting process of rate λ and let {N1(t); t > 0} and {N2(t); t > 0} be two counting processes

constructed as follows. Suppose that each arrival in {N (t); t > 0} is sent to the first process N1(t) with probability p and to the second process N2(t) with probability 1−p. We are therefore

combining a Poisson(λ) process with a Bernoulli(p) process. The resulting processes {N1(t); t >

0} and {N2(t); t > 0} are also two Poisson processes with rates pλ and (1 − p)λ respectively.

A non-homogeneous Poisson process is a Poisson process whose rate λ is non constant over time. A non-homogeneous Poisson process with time varying arrival rate λ(t) is defined as a counting process {N (t); t > 0} that for all t > 0, λ(t) > 0 also satisfies:

P {N(t, t + δt) = 0} = e−Rtt+δλ(u)du ≈ 1 − λ(t)δ + o(δ) P {N(t, t + δt) = 1} = Z t+δ t λ(u)du  e− Rt+δ t λ(u)du ≈ λ(t)δ + o(δ) P {N(t, t + δt) ≥ 2} ≈ o(δ)

Which means that over intervals of a small length (where the variation of λ(t) is small), the number of arrivals in an interval of length δ is still a Poissonr.v. with parameter λ(t)δ, since it is assumed thatRt+δ

t λ(u)du = λ(t)δ. This result is very useful for simulating Poisson processes.

Let us note that for the non-homogeneous Poisson process, all the properties listed above hold, except for the stationarity of the increments. However, using the Mapping theorem reported in Kingman (1992) (page 18), it can be proven that a transformation of the domain of the process allows to rewrite a non-homogeneous Poisson process as a homogeneous Poisson process, with both stationary and independent increments.

Poisson processes are intensively used in queue theory, i.e. the study of waiting lines. Let {N (t); t > 0} be a Poisson process with rate λ. Imagine that upon arrival, every individual is independently assigned an Exponential distribution with rate µ (i.e. the mean lifetime is 1/µ) after which he is served. This is known as an immigration-death process or, in queuing theory, the M/M/∞ queue. Let us now consider {R(t), t > 0} to be the prevalent cases, i.e. the people who have arrived and are waiting to be served. Given the arrival Poisson process, R(t) increases at a rate λ and, given the Exponential distribution of the waiting times, R(t) decreases at a rate ν(t) = r(t)µ, where r(t) is the realization of the arrival process R(t). The service process S(t) inherits the Poisson properties of the arrival process, since, in a small interval of length δ, it can be seen as a sum (convolution) of thinned Poisson processes (those that describe the individuals that arrived at t and waited a time τ before service). It follows that {S(t); t > 0} is a non-homogeneous Poisson process with rate ν(t) = r(t)µ.