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The Poisson process is a stochastic process that models many real-world phenomena. We present the definition of the Poisson process and discuss some facts as well as some related probability distributions. Finally, we give some new applications of the process.

Poisson process is used to model the occurrences of events and the time points at which the events occur in a given time interval, such as the occurrence of natural disasters and the arrival times of customers at a service center. It is named after the French mathematician Siméon Poisson (1781-1840). In this paper, we first give the definition of the Poisson process (Section 2). Then we stated some theroems related to the Poisson process (Section 3). Finally, we give some examples and compute the relevant quantities associated with the process (Section 4).

A Poisson process with parameter (rate)

1)

2)

3)

We give some properties associated with the Poisson process. The proofs can be found in [

Theorem 1 The

Theorem 2 The interarrival times

Theorem 3 Conditioned on

Theorem 4 If

The marked Poisson processes have been applied in some geometric probability area [

1) Suppose the number of calls to a phone number is a Poisson process

For a fixed

2) On average, how many calls arrive when the user is on the phone?

Suppose the user is talking on the

3) In a single server system, customers arrive in a bank according to a Poisson process with parameter

4) Suppose team A and team B are engaging in a sport competition. The points scored by team A follows a Poisson process

Let

5) Given that there are

6) When does a car accident happen? Suppose a street is from west to east and another is from south to north, the two streets intersect at a point

7) Occurrences of natural disasters follow a Poisson process with parameter

where

8) Suppose that

The average time to settle all claims received before

Suppose

where

Clearly,

9) Customers arrive at a shopping mall follows a Poisson process with parameter

Condition on

Then

where

Hence,

That is, the number of customers existing at time

The average number of customers exist at the mall closing time is

10) Customers arriving at a service counter follows a Poisson process with parameter

Condition on

Then

which is the binomial distribution with

That is, the number of customers served longer than

Poisson process is one of the most important tools to model the natural phenomenon. Some important distributions arise from the Poisson process: the Poisson distribution, the exponential distribution and the Gamma distribution. It is also used to build other sophisticated random process.

Kung-Kuen Tse, (2014) Some Applications of the Poisson Process. Applied Mathematics,05,3011-3017. doi: 10.4236/am.2014.519288