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What is Poisson Distribution?

Module by: Lekulana Kolobe. E-mail the author

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Summary: This module introduces Poisson Distribution and gives an example of its application. It also includes an exercise to engage the reader.

Poisson distribution is a "discrete probability distribution. It expresses the probability of a number of events occurring in a fixed time if these events occur with a known average rate, and are independent of the time since the last event" [Wiki]. Such events are said to be memoryless.

Most queuing systems' characteristics such as arrival and departure processes are described by a poisson distributions. Assuming that arrivals and departures are random and independent i.e. they exhibit pure-chance property; arrivals are described by a poisson random variable or poisson random distribution as shown by equation 1.1 [Wiki, Char]

The probability that there are exactly k occurrences (k being a non-negative integer, k = 0, 1, 2, ...) is

Figure 1
Equation 1.1
Equation 1.1 (clip_image001.gif)

Where

e is the base of the natural logarithm (e = 2.71828...),

k! is the factorial of k,

λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average every 4 minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with λ = 2.5.

Exercise 1

(taken from Bajpai A.C, 1974) The average rate of telephone calls received at an exchange of 8 lines is 6 per minute. Find the probability that a caller is unable to make a connection if this is defined to occur when all lines are engaged within a minute of the time of the call.

Solution

We first need to make an assumption that the overall rate of calls is constant, then we can use equation 1.1 as follows: Since our time unit is 1 minute, then λ = 6

Figure 2
Equation 1.1
Equation 1.1 (clip_image001.gif)
Which leads to
Figure 3
After substitution
After substitution (clip_image003.gif)
The probability of not being able to make a call occurs only when there are at least 9 calls in any interval of a minute.
Figure 4
Summation Equation
Summation Equation (clip_image005.gif)
So, this leds to p(k+1) equals to:
Figure 5
From Summation equation we get
From Summation equation we get (clip_image007.gif)
Solving this bit by bit with k from 0 to 8, we get p(k+1) = 0.8546.

References:

Extracted from " http://en.wikipedia.org/wiki/Poisson_distribution" Last Accessed on 13 March 2006

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