Question

Show that, for a stable $M / M / 1$ queueing system, $$ \operatorname{Var}\left[N_q\right]=\frac{\rho^2\left(1+\rho-\rho^2\right)}{(1-\rho)^2 .} $$

   Show that, for a stable $M / M / 1$ queueing system,
$$
\operatorname{Var}\left[N_q\right]=\frac{\rho^2\left(1+\rho-\rho^2\right)}{(1-\rho)^2 .}
$$

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 4

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Step 1

In this model, $\lambda$ is the arrival rate, $\mu$ is the service rate, and $\rho = \lambda / \mu$ is the traffic intensity. For the system to be stable, it is necessary that $\rho < 1$. Show more…

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Show that, for a stable $M / M / 1$ queueing system, $$ \operatorname{Var}\left[N_q\right]=\frac{\rho^2\left(1+\rho-\rho^2\right)}{(1-\rho)^2 .} $$

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