Consider a queueing system where customers arrive according to a Poisson process with rate $\lambda$, but the service facility consists of two parallel servers. A customer upon entry into the service facility will proceed to either server 1 with probability 0.25 or to server 2 with probability 0.75. While the customer in the service facility is receiving his/her service, no other customer is allowed into the service facility. If the service rates of these two servers are exponentially distributed with rate $\mu_i$ (i = 1, 2), calculate the mean waiting time of a customer in this queueing system.

My approach is to look at this system as an M/G/1 with variable service times, since, from the point of view of a general customer in queue, there are Poisson arrival with rate $\lambda$ in the system, and the departure rates are of rate $\mu_1$ with probability 0.25 and $\mu_2$ with probability 0.75, so we can use the Polaczek-Khinchin formula:

$W = \frac{\lambda \bar {x^2}}{2(1-\rho)}$

where we have $\mu = \mu_1/4 + 3\mu_2/4$, $\rho = \lambda\mu$, but I have some doubts on calculate $\bar x$ and $\bar {x^2}$, since my new departure rate $\mu$ now is linear combination of two dependent random variables with exponential distribution, and so I don't know how to treat it. Can someone help me, please? Thanks



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