I need help with one of the questions in my lecture notes.
Consider two parallel machines with a common buffer where jobs arrive according to a Poisson stream with rate $λ$. The processing times are exponentially distributed with mean $\frac{1}{µ_1}$ on machine 1 and $\frac{1}{µ_2}$ on machine 2 (µ1 > µ2). Jobs are processed in order of arrival. A job arriving when both machines are idle is assigned to the fast machine. We assume that
$\rho = \frac{\lambda}{\mu_1 + \mu_2} < 1$.
So I thought I should just set up the global balance equations, but I'm pretty much stuck after.
Using the notation that $\mu = \mu_1 + \mu_2$ and state $1f$ for the fast machine and $1s$ for the slow machine ($p_1 = p_{1f} + p_{1s}$), we obtain for the balance equations:
$\lambda p_0 = \mu_1 p_{1f} + \mu_2 p_{1s}$
$(\lambda + \mu_1) p_{1f} = \lambda p_0 + \mu_2 p_2$
$(\lambda + \mu_2) p_{1s} = \mu_1 p_2 $
$(\mu_1 + \mu_2 + \lambda) p_n = \lambda p_{n-1} + \mu p_{n+1}$, for $n \geq 2$
I imagine the result for $p_n$, $n \geq 2$ will be similar to the standard M/M/1 queue, i.e. $p_n = \rho^{n-1} p_1$, but I really don't know how to obtain this.