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having a bit of trouble with the following question:

Consider a two-server queue with Exponential arrival rate $\lambda$. Suppose servers 1 and 2 have exponential rates $\mu_{1}$ and $\mu_{2}$, with $\mu_{1}$ > $\mu_{2}$. If server 1 becomes idle, then the customer being served by server 2 switches to server 1.

a) Identify a condition on $\lambda, \mu_{1}, \mu_{2}$ for this system to be stable, i.e. queue not infinitely long

b) Using that condition, find long-run proportion of time that server 2 is busy

What I have so far is for a system to be stable it must have p $\le$ 1. How this ties into the second part I'm not sure about. Any help appreciated, thanks!

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This is how I see this problem

Since $\rho = \frac{E(S)}{E(X)} < 1$ we just need to find an expressioin for E(X) and E(S) in terms of $\mu$ and $\lambda$. Clearly E(X) is just $\lambda$.

We know that E(S) is the expected departure rate of individuals from the system. In the case of a regular M/M/s queue with each server having rate $\mu$, this would be 1/$s\mu$ since the departure rate of individuals would be the minimum of the s exp($\mu$) servers.

An individual departs the system in one of two ways:

  1. S1 (server 1) finishes before S2. Then the individual from S2 would move to S1 and a new individual would join S2.
  2. S2 finishes before S1 and a new individual joins S2.

Clearly the departure rate is the minimum of two exponential RVs and its expectation would be $\frac{1}{\mu_1 + \mu_2}$

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