There are two car M/M/1 queues Q1 and Q2. Arrival rate of Red car and Green car in Q1 is $\lambda_{1R}$ and $\lambda_{1G}$ respectively. Similarly arrival rate of red car and green car in Q2 is $\lambda_{2R}$ and $\lambda_{2G}$ respectively. There are another two service M/M/1 service queues for Red car and Green car. The RED queue is for servicing Red car with service rate $\mu_R$ and GREEN queue is for servicing Green car with service rate $\mu_G$. One car will leave the queue Q1/Q2 if and only if the car just ahead of it from the same queue leaves RED/GREEN service station according to it's color. Given that arrival of cars to Q1 and Q2 is a Poisson process and any queue length can be infinite, what is the average waiting time for servicing Red car and Green car? [note: I post the picture for clear understanding]. I worked on the problem to get the average waiting time but not sure whether it's correct.
$\rho_R = \frac{\lambda_{1R}+\lambda_{2R}}{\mu_R}$; $\rho_G = \frac{\lambda_{1G}+\lambda_{2G}}{\mu_G}$
using Little's theorem and superposition theorem of Poisson process average waiting time can be calculated as,
$W = \frac{\frac{\rho_R}{1-\rho_R}+\frac{\rho_G}{1-\rho_G}}{\lambda_{1R}+\lambda_{2R}+\lambda_{1G}+\lambda_{2G}}$
