Question

Show that for a steady state $M / M / 1$ queueing system, $$ \sigma_q^2=\frac{(2-\rho) \rho W_s^2}{(1-\rho)^2} . $$

   Show that for a steady state $M / M / 1$ queueing system,
$$
\sigma_q^2=\frac{(2-\rho) \rho W_s^2}{(1-\rho)^2} .
$$

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 8

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Step 1

In an $M/M/1$ queue, arrivals follow a Poisson process with rate $\lambda$, and service times are exponentially distributed with rate $\mu$. The traffic intensity of the system is given by $\rho = \frac{\lambda}{\mu}$. Show more…

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Show that for a steady state $M / M / 1$ queueing system, $$ \sigma_q^2=\frac{(2-\rho) \rho W_s^2}{(1-\rho)^2} . $$

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