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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?

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    $\begingroup$ Yes, it is correct! Now you can write all the state probabilities in terms of $p_0$ and equate the sum to 1 to get $p_0$. $\endgroup$ – Submartingale Nov 17 '13 at 17:07

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