What is Poisson Distribution? 1.1 2006/02/15 05:06:08.441 US/Central 2006/02/16 02:12:18.125 US/Central Lekulana Emmanuel Kolobe klekulana@hotmail.com Lekulana Emmanuel Kolobe klekulana@hotmail.com Poisson Distribution Queuing systems 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

Equation 1.1

Wheree 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. (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. 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

Equation 1.1

Which leads to

After substitution

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.

Summation Equation

So, this leds to p(k+1) equals to:

From Summation equation we get

Solving this bit by bit with k from 0 to 8, we get p(k+1) = 0.8546.

References: Characteristics of Queuing Systems. accessed on 14-02-2006 at http://www.bsbpa.umkc.edu/classes/ashley/Chaptr14/sld007.htm Bajpai A.C., Mustoe L.R., and Walker D. Engineering Mathematics. John Willey. 1974. Pages 678 to 683 Extracted from " http://en.wikipedia.org/wiki/Poisson_distribution" Last Accessed on 13 March 2006