Convolution Method

• The probability distribution of a sum of two or more independent random variables is called a convolution of the distributions of the original variables.

• Erlang distribution.
  • an Erlang random variable X with parameters $(K, \theta)$ can be shown to be the sum of K independent exponential random variables $X_i ~~ (i = 1,...,K)$, each having a mean $1/K \theta$
    
    $$X = \sum_{i=1}^K X_i$$
    
    

  • Using equation (9.3) that can generate exponential variable, an Erlang variate can be generated by
    
    $$\begin{array}{ll} X & = \sum_{i=1}^K \frac{-1}{K\theta} \ln ... ...{K\theta} \ln \left( \prod_{i=1}^K R_i \right)\\ \end{array}$$
    
    

• Example 9.9 on page 343