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.

  • $\begingroup$ Looks like you're missing one equation: $(\lambda+\mu_1+\mu_2)p_2 = \lambda p_{1s}+\lambda p_{1f}+\mu p_3$. $\endgroup$ – Math1000 May 18 at 22:22
  • $\begingroup$ See the solution to exercise 21 on page 145 of this text: win.tue.nl/~iadan/queueing.pdf $\endgroup$ – Math1000 May 18 at 22:47

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