Consider the M$/$G$/1$ model with $\backslash$lambda $=0\mathrm{.}2$ and $\backslash$mu $=0\mathrm{.}25.$

T$($a$)$ Use the Excel template for this model $($or hand calculations$)$ to find the main measures of performance$$L$,$ Lq$,$ W$,$ Wq$$for each of the following values of $\backslash$sigma : $4,3,2,1,0.$

$($b$)$ What is the ratio of Lq with $\backslash$sigma $=4$ to Lq with $\backslash$sigma $=0?$ What does this say about the importance of reducing the variability of the service times?

$($c$)$ Calculate the reduction in Lq when $\backslash$sigma is reduced from $4$ to $3,$ from $3$ to $2,$ from $2$ to $1,$ and from $1$ to $0.$ Which is the largest reduction? Which is the smallest?

$($d$)$ Use trial and error with the template to see approximately how much $\backslash$mu would need to be increased with $\backslash$sigma $=4$ to achieve the same Lq as with $\backslash$mu $=0\mathrm{.}25$ and $\backslash$sigma $=0.$

Please answer these questions thoruoghly!