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Operations Research Models and Methods
Computation Section
Subunit Continuous Distributions
 - Exponential

The Exponential distribution is often used to model situations involving the random time between arrivals to a service facility. When the average arrival rate is and the arrivals occur independently, then the time between arrivals has an Exponential distribution characterized by the single positive parameter . The density has the shape shown in the figure.

Integrating the density yields a closed form expression for the CDF. The distribution has equal mean and standard deviation, and the skewness is always 4. Because it is skewed to the right, the mean is always greater than the median. The probability that the random variable is less than its mean is independent of the value of .



The Exponential distribution is the only memoryless distribution. If an event with an Exponential distribution with parameter has not occurred prior to some time, the distribution of the time until it does occur has an Exponential distribution with the same parameter.

The Exponential distribution is intimately linked with the discrete Poisson distribution. When the time between arrivals of some process is governed by the exponential distribution with rate , the number of arrivals in a fixed interval T is governed by the Poisson distribution with mean value T. The arrival process is called a Poisson process.



Example: Telephone calls arrive at a switchboard every 30 seconds on average. Because callers are independent, the calls arrive at random. We would like to know the probability that the time between one call and the next is greater than 1 minute?

The average arrival rate is 2 per minute. The add-in computes the probability that the random variable is less than 1. The required probability is then is

P(x > 1) = 1 – 0.865 = 0.135.

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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved

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