Chapter 12: The Poisson
process
CIS 3033
12.1 Random points
Random process: not just one or two random
variables that play a role but a whole collection.
Poisson process models the most random way to
distribute points in time or space.
The Poisson process: a very large population, and
each member of the population has a very small
probability to produce a point of the process.
12.2 Random arrivals
Example: Telephone calls arrive at random times
X1,X2, . . . at the telephone exchange during a time
interval [0, t].
Basic assumptions:
• Independence: The numbers of arrivals in disjoint
time intervals are independent random variables.
• Homogeneity: The rate λ at which arrivals occur is
constant over time.
Homogeneity implies weak stationarity: in a
subinterval of length t, the expectation of the number
of telephone calls is λt, i.e., E[Nt] = λt.
12.2 Random arrivals
When the [0, t] interval is evenly divided into n
subintervals, the expected number of calls in each
subinterval is λt/n.
When n is large enough, each subinterval contains at
most one call, so can be seen as a random variable
Ber(λt/n).
The total number of calls Nt has a Bin(n, λt/n)
distribution. When n goes to infinite, the distribution
becomes Pois(λt).
12.2 Random arrivals
Let X have a Poisson distribution with parameter μ;
then E[X] = μ and Var(X) = μ.
12.2 Random arrivals
12.3 One-dimensional Poisson process
The previous result is about the number of phone
calls, and the following is about their timing.
The one-dimensional Poisson process with intensity
λ is a sequence X1, X2, X3, . . . of random variables
having the property that the interarrival times X1,
X2 − X1, X3 − X2, . . . are independent random
variables, and each with an Exp(λ) distribution and
expectation 1/λ.
The single-server queue example in Chapter 6.
12.3 One-dimensional Poisson process
For i = 1, 2, . . . the arrival time Xi is the sum of i
interarrival times, so has a Gam(i, λ) distribution.
From Chapter 11:
12.3 One-dimensional Poisson process
Given that the Poisson process has n points in
the interval [a, b], the locations of these points
are independently distributed, each with a
uniform distribution on [a, b].
A similar analysis can be carried out for higherdimensional Poisson processes, as explained in
Section 12.4.