1
$\begingroup$

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!

$\endgroup$
0
$\begingroup$

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 $\frac{1}{\mu_1 + \mu_2}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.