# Queueing Theory: Fast and a slow machine

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.

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