Poisson Process

The Poisson process N(t) is an example of a counting process, that is N(t) is the number of times something has happened up to time t.

Example: In our ATM machine example let N(t) be the number of customers served by the ATM machine at time t

Because of the way it is defined every counting process has the following properties:
1)N(t) 0
2) N(t) is an integer
3) If s <t then N(s)N(t)
4) If s<t then N(t)-N(s) is the number of events that have occured in the interval (s,t).

5) A counting process is said to have independent increments if the number of events that occur in disjoint intervals are independent.
6) A counting process is said to have stationary increments if the distribution of the number of events that occur in any interval of time depend only on the length of the interval.

Example: The process of our ATM machine probably has independent increments but not stationary increments. Why?

The most important example of a counting process is the Poisson process. To define it we need the following notation, called Landau's o symbol:
a function f is said to be o(h) if lim_h0f(h)/h = 0

Example: f(x)=x² is o(h) but f(x)=x is not.

A counting process N(t) is said to be a Poisson process with rate l>0 if
1) N(0)=0
2) N(t) has stationary and independent increments
3) P(N(h)=1) = lh + o(h)
3) P(N(h)2) = o(h)

Notice that this implies that during a short time period the probability of an event occuring is proportional to the length of the interval and the probability of a second (or even more) events occuring is very small.
It can be shown that this implies the follwoing:
P(N(t+s)-N(s)=n) = dpois(n,lt)
The proof is straitforward: note that we can subdivide the interval (s,t) into many (say N) subintervals of length h. Then if the r.v. Y_i is the number of events in the i^th subinterval Y_i is a Bernoulli r.v. because there are supposed to be either none or maybe one event in any small interval. Now N(t+s)-N(s) = SY_i and the conclusion then follows from the Poisson approximation to the Binomial.

Here are some of the main results for Poisson processes:

Interarrival Times and Waiting Times

Let T₁ be the time when the first event occurs, T₂ the time from the first event until the second event etc. The sequence T₁, T₂, .. is called the sequence of interarrival times.
Note that {T₁>t} is equivalent to {no events occured in [0,t]} and so
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and we see that T₁ ~ Exp(1/l). But
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because of independent and stationary increments. So we find that T₂ ~ Exp(1/l) and that T₁

T₂. By induction it is clear that the sequence {T_n,n=1,2,..} is an iid sequence of exponential r.v. with mean l.

Let S_n be the time elapsed until the n^th event occurs. Clearly S_n = Sⁿ_i=1T_i, and so we find S_n ~ G(n,l).

Example: Let N(t) be the number of cars that passed a certain intersection where we start counting at 10am. Say the mean number of cars passing per second is 0.65. What is the probability that the 1000^th car arrives before 10:10am?
We have P(S₁₀₀<600) = pgamma(1000,600,0.65) = 0.9772