Hyperexponential distribution

Properties

Since the expected value of a sum is the sum of the expected values, the expected value of a hyperexponential random variable can be shown as

: E[X] = \int_{-\infty}^\infty x f(x) , dx= \sum_{i=1}^n p_i\int_0^\infty x\lambda_i e^{-\lambda_ix} , dx = \sum_{i=1}^n \frac{p_i}{\lambda_i}

and

: E!\left[X^2\right] = \int_{-\infty}^\infty x^2 f(x) , dx = \sum_{i=1}^n p_i\int_0^\infty x^2\lambda_i e^{-\lambda_ix} , dx = \sum_{i=1}^n \frac{2}{\lambda_i^2}p_i,

from which we can derive the variance:

:\operatorname{Var}[X] = E!\left[X^2\right] - E!\left[X\right]^2 = \sum_{i=1}^n \frac{2}{\lambda_i^2}p_i - \left[\sum_{i=1}^n \frac{p_i}{\lambda_i}\right]^2 = \left[\sum_{i=1}^n \frac{p_i}{\lambda_i}\right]^2 + \sum_{i=1}^n \sum_{j=1}^n p_i p_j \left(\frac{1}{\lambda_i} - \frac{1}{\lambda_j} \right)^2.

The standard deviation exceeds the mean in general (except for the degenerate case of all the *λ*s being equal), so the coefficient of variation is greater than 1.

The moment-generating function is given by

:E!\left[e^{tx}\right] = \int_{-\infty}^\infty e^{tx} f(x) , dx= \sum_{i=1}^n p_i \int_0^\infty e^{tx}\lambda_i e^{-\lambda_i x} , dx = \sum_{i=1}^n \frac{\lambda_i}{\lambda_i - t}p_i.

Fitting

A given probability distribution, including a heavy-tailed distribution, can be approximated by a hyperexponential distribution by fitting recursively to different time scales using Prony's method.

References

References

1. (2007). "Estimation of the Hyperexponential Density with Applications in Sensor Networks". International Journal of Distributed Sensor Networks.
2. (1993). "Queueing Theory in Manufacturing Systems Analysis and Design". Springer.
3. (1998). "Fitting mixtures of exponentials to long-tail distributions to analyze network performance models". Performance Evaluation.