Expected value of OR gate

Question

Suppose a system such as A+B where + means OR. Suppose A and B are identical. Each event obeys exponential distribution with mean $\lambda$.

What is the mean $\lambda_{\text{System}}$ for the whole system?

Answer 1

Here is the interpretation of the question that supports the answer in Did's comment.

Suppose that $A$ and $B$ are two logical variables (inputs to an OR gate) that have value $0$ at time $t=0$ and change to value $1$ at times $t_1=X$ and $t_2=Y$ where $X$ and $Y$ are modeled as independent exponential random variables with parameters $\theta_X$ and $\theta_Y$ respectively, and thus means $\frac{1}{\theta_X}=\lambda_X$ and $\frac{1}{\theta_Y}=\lambda_Y$ respectively. If $C = A\vee B$ is the output of the OR gate, then $C$ also has initial value $0$ at $t=0$ and changes to $1$ at time $Z = \min\{X, Y\}=\min\{t_1,t_2\}$. It is straightforward to verify that $Z$ is also an exponential random variable whose parameter is $\theta_X+\theta_Y$. The expected value of $Z$ is $$E[Z] = \frac{1}{\theta_X+\theta_Y} = \frac{1}{\frac{1}{\lambda_X}+\frac{1}{\lambda_Y}} = \frac{\lambda_X\lambda_Y}{\lambda_X+\lambda_Y}.$$