Prove that, for the M / G / 1 queueing system, ^2= (λ^3 E[s^3])/(3(1-ρ))+((λ^2 E[s^2])/(2(1

Question

Prove that, for the M / G / 1 queueing system,  ^2=     (λ^3 E[s^3])/(3(1-ρ))+((λ^2 E[s^2])/(2(1

Accepted Answer

Step 1: Understand the $M/G/1$ queueing system.
The $M/G/1$ queueing system is a single-server q

Prove that, for the $M / G / 1$ queueing system, $$ \begin{aligned} \sigma_N^2= & \frac{\lambda^3 E\left[s^3\right]}{3(1-\rho)}+\left(\frac{\lambda^2 E\left[s^2\right]}{2(1-\rho)}\right)^2 \\ & +\frac{\lambda^2(3-2 \rho) E\left[s^2\right]}{2(1-\rho)}+\rho(1-\rho) \end{aligned} $$