# Multiclass Markov process

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,

$W = \frac{\frac{\rho_R}{1-\rho_R}+\frac{\rho_G}{1-\rho_G}}{\lambda_{1R}+\lambda_{2R}+\lambda_{1G}+\lambda_{2G}}$

• Are the service parameters for red and green cars in the first services (mixed/left) the same as in the case of the second services (separated/right)? – zoli Mar 1 '15 at 22:04
• zoli, service parameters for Q1 and Q2 are not given. It's like two stream of cars are coming from two different traffic and accessing shared service station "Red" and "Green". Service rate for Red and Green station shows how much time one red car and green takes for servicing. – marcella Mar 1 '15 at 22:49
• ${\rho}_R$ and $\rho_G$ are OK. Little's theorem sais for the average waiting time that it equals $\frac{1/{\mu_X}}{1-{\rho_X}}$, $X=R,G$. So, we have the average waiting time for both kinds of cars. How did you get your $W$? – zoli Mar 1 '15 at 23:29
• I just edited my question. Please have a look. Thanks for your interest. – marcella Mar 2 '15 at 0:08
• Sorry, I give up. I have to interpret your instructions regarding Q1/Q2 queues. I cannot do that. I don't see the $\mu$ parameters for them. So I doubt that Q1/Q2 are M/M/1. But I may be mistaken. – zoli Mar 2 '15 at 0:42