Question

Show that, for the GI/M/1 queueing system, $$ \sigma_{N_q}^2=\frac{\rho\left(1-\pi_0\right)\left[2-\pi_0-\rho\left(1-\pi_0\right)\right]}{\pi_0^2} . $$

   Show that, for the GI/M/1 queueing system,
$$
\sigma_{N_q}^2=\frac{\rho\left(1-\pi_0\right)\left[2-\pi_0-\rho\left(1-\pi_0\right)\right]}{\pi_0^2} .
$$

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 65

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Step 1

In this model, the interarrival times are generally distributed (GI), the service times are exponentially distributed (M), and there is a single server (1). The system's state can be described by the number of customers in the system, \( N(t) \), at time \( t \). Show more…

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Show that, for the GI/M/1 queueing system, $$ \sigma_{N_q}^2=\frac{\rho\left(1-\pi_0\right)\left[2-\pi_0-\rho\left(1-\pi_0\right)\right]}{\pi_0^2} . $$

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