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Poisson Process

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A Poisson process is a random process that counts the number of events occurring and the time interval within which the events have occurred . The time between the two consecutive or back to back events is exponentially distributed which is denoted by ‘λ’ and the time between each of these consecutive events is independent of the time of other events. It happens consecutively hence it is a continuous time process. This was Poisson process definition.
A Poisson process is a continuous process which goes on with time and has the following properties.
1. N(0) = 0 (it means that the initial value should be 0 or the process should start at 0).
2. The number of times the event has occurred should be independent of others.
3. The distribution of the number of events that have occurred during any interval only depends on the duration of that interval.
4. The occurrence of any two or more events should not be simultaneous.

Poisson process definition results in the following conclusions:
1. The Probability distribution of the events is a Poisson distribution.
2. The distribution of the waiting time is exponentially distributed.
3. The occurrence of any event is distributed uniformly on any time interval.

Now we shall study the different types of the Poisson processes.
1. Homogeneous Poisson process : this process is known by the rate parameter ‘λ’ which is also sometimes known as intensity. It is distributed in a manner that the number of events during a time interval (t + tR) follows a Poisson process.
The relation is given by
P [N(t + R) - N(t)) = k] = e^[(-λR)(λR)^k] / k! k=(0,1,2,3,,,,… n)
Here [N(t + R) - N(t)) = k] is the number of events that occur in the time interval(t + tR)
So we can see that that the homogeneous Poisson process is known by the rate parameter ‘λ’, just like the normal Poisson process is known by the ‘λ’ which is the number of events that will occur in the unit amount of time.
2. Non homogeneous Poisson process: in non homogeneous Poisson process the interval of time between the two events is not regular or equal and may change over time. The general equation of the non homogeneous Poisson process is denoted by
λ (p,q) = ∫ λ(t) dt
where the ‘λ’ is the rate parameter and the lower limit and the upper limit of the integral is ‘p’ and ‘q’.
The number of arrivals in the time intervals is given by :
P[n(q) - n(p) = k] = [e^(-λ(p,q))λ(p,q)^k]/k! wher k =(0,1,2…. n)
Spatial Poisson process: the spatial Poisson process is a little different from the normal time based Poisson process as it is only one dimensional or it depends on only one line. Here only the index variable changes from the standard variable of the normal Poisson process. There is a need of the spatial Poisson process when the random variables count the number of events.
Space- time Poisson process: this process is exclusively for the space-time Poisson process and it possesses a separate variable for space and time. Though both the dimensions can be studied within the same Poisson process but since both the dimensions possess different properties hence it is easy to study them differently.
The general formula of the space-time Poisson process is
λs(t) = ∫λ(x,t) dμ (x)
here, the lower limit of the integral is ‘s’ which is any spatial region.
The general condition of any process to behave like a Poisson process is
lim∆t→0 P[n(t + ∆t) - n(t) > 1| n(t + ∆t) – n(t) > 1)] = 0
which simply means that the events won’t come simultaneously.
Separable space time process: the special case case in this type can be defined as
λ(x,t) = f(x) λ(t)
lets assume that f(x) is the spatial Poisson process. So the evaluation of this spatial Poisson process is similar to the evaluation of the Poisson process with rate function ‘R,.
Now applications of Poisson process are discussed:
1. The number of goals in a football match are a Poisson process.
2. The arrival and the departure of the customers in any queue is a Poisson process.
3. During rain, the number of the raindrops dropped in a specific area can be calculated by the Poisson process.
4. In the switchboard, there are several telephone calls, which may be incoming calls or outgoing calls, existing at the same time. To compute the telephone calls coming at the switching board the Poisson process can be used.
5. The number of particles which would emit when there is an radioactive process can be determined by an application of Poisson process. In this case we may use non homogeneous Poisson process because the emission of the radioactive substance may decline as the particles are emitted.
6. There are several requests of an individual page on any web server which can be managed by the Poisson process.

Applications of Poisson Process

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Poisson process can be defined as process which is used to count the total number of occurring and occurred events. Also Poisson Process is used to calculate that time interval which is taken by events to occur. The time interval has an exponential (power) distribution which occurs between each couple of alternative events and each inter arrival time intervals is supposed to be independent as compared to other arrival time intervals. Poisson process, which is a continuous-time counting process, is considered a fine model for studying the decay in radioactive elements. Widely, Poisson concept is used in study of Probability.
Basic form of Poisson process is a continuous-time counting process. Mathematically, M (t), t ≥ 0. This process has following characteristics:

1) N (0) = 0 that is at time interval zero, total counting of events are zero.

2) There are independent increments in number of events that is Numbers of happening calculated in disjoint intervals are completely independent from each other.

3) Probability distribution of number of counted events in any time interval only depends on length of interval and it is referred as stationary increments and also no counted occurrences are simultaneous.
Stuttering Poisson process occurs when counted occurrences are alternate that is probability of one or more events occur at a very less time interval is non-zero.

Let’s see Applications of Poisson Process.
Let’s consider that there are X claims which are being paid to the clients of a pharmacy company. These payments consist of some distribution and happen at arrival times of counting process. If counting process is a Poisson process and claims are considered to be ‘i’ random variables. Then total amount of claims is a Poisson process. It will be mathematically expressed as:
At = ∑ Mt X m,
Here random variables are also independent from counting process.