Question

Show that, for an $M / M / c$ queueing system in the steady state, $$ \sigma_{N_q}^2=\frac{\rho C[c, a]\{1+\rho-\rho C[c, a]\}}{(1-\rho)^2}, $$ and $$ \sigma_N^2=\sigma_{N q}^2+a(1+C[c, a]) . $$

   Show that, for an $M / M / c$ queueing system in the steady state,
$$
\sigma_{N_q}^2=\frac{\rho C[c, a]\{1+\rho-\rho C[c, a]\}}{(1-\rho)^2},
$$
and
$$
\sigma_N^2=\sigma_{N q}^2+a(1+C[c, a]) .
$$

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen 2nd Edition

Chapter 5, Problem 17

Instant Answer

Step 1

In an $M/M/c$ queueing system: - $\lambda$ is the arrival rate, - $\mu$ is the service rate per server, - $c$ is the number of servers, - $\rho = \frac{\lambda}{c\mu}$ is the traffic intensity, - $N_q$ is the number of customers in the queue, - $N$ is the total Show more…

Show all steps

lock

Thanks for your feedback!

Show that, for an $M / M / c$ queueing system in the steady state, $$ \sigma_{N_q}^2=\frac{\rho C[c, a]\{1+\rho-\rho C[c, a]\}}{(1-\rho)^2}, $$ and $$ \sigma_N^2=\sigma_{N q}^2+a(1+C[c, a]) . $$

Play audio

Feedback

Powered by NumerAI

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later