DES - Definition of 'Superposition'

Superposition:

Let $X_1$, $X_2$, …, $X_n$ be independent random variables, each having an exponential distribution:

$$X_i\sim Exp\left(\frac{1}{{\lambda }_i}\right)i=1,2,...,n$$

Define the superposition of $X_1$, $X_2$, …, $X_n$ as

$$X=\underset{i=1,...,n}{\min }{X}_i$$

Also we have that the superposition in characterised by an exponential distribution of rate $\Lambda$, where:

$$\Lambda =\sum _{i=1}^n{X}_i$$

• Demonstration

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~Example: Why is the superposition useful?

Suppose we have a Poisson Clock Structure with two events $X$ and $Y$ declared as:

$$\begin{array}{r}X\sim Exp\left(\frac{1}{\lambda }\right)\\ Y\sim Exp\left(\frac{1}{\mu }\right)\end{array}$$

Then the following equations holds:

$$\begin{array}{rlr}P(X<Y+t)=& 1-\frac{\mu }{\lambda +\mu }{e}^{-\lambda t}& ,t\ge 0\\ =& \frac{\lambda }{\lambda +\mu }{e}^{-\lambda t}& ,t\ge 0\end{array}$$

• Demonstration

And for $t=0$:

$$P(X<Y)=\frac{\lambda }{\Lambda }$$

Where: $\Lambda$ is the rate of the distribution of the superposition, which is defined in this case as:

$$\Lambda =\lambda +\mu$$

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