I have a problem using M/M/2 queue theory. In this case, the service rate $\mu$ varies with constant $\lambda$. For example : between state 0 and 1, I use service rate $\mu$ but for other states I use service rate $2\mu$.
Between state 0 and 1:
$\lambda p_0 = \mu p_1$
$p_1 = \frac{\lambda}{\mu}p_0=\rho p_0$, for n=1
Other states :
$\lambda p_n = 2\mu p_{n+1}$
$p_{n+1} = \frac{\lambda}{2\mu}p_n=\frac{\rho}{2} p_n$, for n > 1
Is these formula correct?