Average time from two exponential distributed arrival times Suppose that the time between request arrivals at the first server is random and follows an exponential distribution with $\lambda=u_{1}$. Similarly, the time between request arrivals at the second server is also random and follows an exponential distribution with $\lambda=u_{2}$. The first server has probability $p_{1}$ of referring any request it receives to the main server, and the second server has probability $p_{2}$ of doing so. What is the average time between requests referred to the main server?
This was a MCQ and I think by exclusion the answer is $\frac{1}{u_{1} p_{1}+u_{2} p_{2}}$ but I am not sure how to derive it.
 A: You should think about this problem in terms of splitting Poisson processes.
The number of requests received by the first and second server which get referred to the main server are $\text{Poisson}(p_1 \mu_1)$ and $\text{Poisson}(p_2 \mu_2)$, respectively. This implies the number of requests received by the main server is $\text{Poisson}(p_1\mu_1+p_2\mu_2)$ whose interarrival times are $\text{Exp}(p_1\mu_1+p_2\mu_2)$ and have mean $\frac{1}{p_1\mu_1+p_2\mu_2}$
A: Assuming everything is independent:

*

*the first server is receiving requests at a rate of $u_1$ and passing on the proportion $p_1$, so is passing on requests to the central server at a rate of $u_1p_1$


*the second server is receiving requests at a rate of $u_2$ and passing on the proportion $p_2$, so is passing on requests to the central server at a rate of $u_2p_2$


*so the central server is receiving requests at a rate of  $u_1p_1+u_2p_2$, and given the memoryless property of the exponential distribution these are exponentially distributed, making the expected time between arrivals $\frac{1}{u_1p_1+u_2p_2}$
