I am working on traffic signals for a city transport system. I modeled the city transport using a queuing network as shown in the following image
Arrival rate of "A" cars from outside is S1 and arrival rate of "B" cars is S2. "A" and "B" cars can not enter into traffic-3 and traffic-1 respectively.
Traffic 1,2,3 are modeled as M/M/1-FCFS queue with infinite queue capacity.$(1-P_1)$ and $(1-P_3)$ are exit probabilities for "A" and "B" cars. $\mu_i$ is the service rate of $i_{th}$ traffic. It is given that
$\sum_{i=1}^{3}\mu_i=Constant$.
So,overall arrival rate can be written as
$\lambda1=S1 + P_1.\lambda1$
$\lambda2=S2 + P_2.\lambda2$
Is it possible to find $\mu=\{\mu_1,\mu_2,\mu_3\}$ which will minimize the mean queue length of all three traffics. Can I use Jackson's theorem in this model?