< List of probability distributions

The **hyperexponential distribution** (also called a *mixed exponential *or *parallel m-phase exponential distribution*) is an extended version of the exponential distribution that is created by a probability mixture of exponential distributions. It is used to model systems with mixed failure rates, such as communication networks, production lines, and chemical processes. In this article, we will discuss the concept of the hyperexponential distribution, its properties, applications, and how to calculate its parameters.

## Hyperexponential distribution definition

The hyperexponential distribution is a probability distribution that results from a mixture of exponential distributions with different failure rates. The monikier *hyper*exponential comes from the fact that the distribution’s coefficient of variation is always greater than one [1].

. It is a generalization of the Erlang distribution, which is made by adding exponential distributions with the same failure rate. The hyperexponential distribution is a special case of the phase-type distribution, formed by a convolution or mixture of exponential distributions [2].

## Properties

The hyperexponential distribution is a light-tailed distribution with thin tails [3]. The distribution can have two or more parameters, which correspond to the failure rates of the exponential distributions and the probabilities of their occurrence. The expected value and the variance of the hyperexponential distribution can be calculated analytically. The distribution is also memoryless, which means that the probability of failure in the future is independent of the history of the system.

The probability density function (PDF) is defined as

Where:

- Y
_{i}= an exponentially distributed random variable with rate parameter i. - p
_{i}= probability that X will have an exponential distribution with rate λ.

Generally speaking, the hyperexponential distribution is monotonically decreasing with light tails. This means that the PDF decreases exponentially for large x-values.

## Applications of the hyperexponential distribution

The hyperexponential distribution is commonly used to model systems with mixed failure rates, such as communication networks, production lines, and chemical processes. In communication networks, the hyperexponential distribution is used to model the packet arrival and departure times. In production lines, it is used to model the breakdown and repair times of machines. In chemical processes, it is used to model the lifetimes of catalysts and reactors.

To calculate the parameters, we need to estimate the failure rates of the exponential distributions and the probabilities of their occurrence. This can be done using the maximum likelihood estimation method or the least-squares estimation method. Once the parameters are estimated, we can use them to calculate the expected value, the variance, and other properties of the distribution.

Hyperexponential distributions can have multiple stages. For each stage, parameters must be specified defining probabilities at each stage.

For example, a two-stage distribution, created from two exponential distributions, has three parameters: p, λ_{1} and λ_{2} [5]:

- p: the probability of receiving service at rate λ
_{1}. - p – 1: the probability of receiving service at rate λ
_{2}

## Advantages and Limitations of the Hyperexponential Distribution

The advantages of the hyperexponential distribution include its ability to model systems with mixed failure rates, its analytically calculable expected value and variance, and its memoryless property. The limitations of the hyperexponential distribution include its light tails, which makes it unsuitable for some applications, and its assumption of exponential failure rates, which may not hold for some systems.

In conclusion, the hyperexponential distribution is a probability distribution that results from a mixture of exponential distributions with different failure rates. It is used to model systems with mixed failure rates, such as communication networks, production lines, and chemical processes. The hyperexponential distribution has several properties, including a light tail, memorylessness, and an analytically calculable expected value and variance. The parameters of the hyperexponential distribution can be estimated using the maximum likelihood estimation method or the least-squares estimation method. Finally, the hyperexponential distribution has advantages and limitations that should be taken into account when choosing a distribution to model a system.

## References

[1] (2013) Hyperexponential Distribution. In: Gass S.I., Fu M.C. (eds) Encyclopedia of Operations Research and Management Science. Springer, Boston, MA.

[2] Package ‘sdprisk’.

[3] Nair, J. et al. The fundamentals of heavy tails: properties, emergence and estimation. Cambridge University Press; New edition (September 29, 2022)

[4] CS 547 Lecture 14: Other Service Time Distributions

[5] Image: Gareth Jones, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons