$1.($Waiting Line Models$)$ A single public health worker is inoculating children against measles at a local clinic. An average of $6$ children per hour is expected to arrive according to a Poisson distribution. He serves the children at a rate of $8$ per hour. Assume that the service$-$time distribution is approximately exponential. Use the Excel "Lec $\backslash (\backslash$mathbf$\{10\}\backslash )$ MMk Model" to answer the following questions.

$($a$)$ For the 'inoculation system', find:

i$)$ the utilization of the health worker

ii$)$ the average time spent by a child in the system

iii$)$ the average number of children in the system

iv$)$ the average number of children in line

v$)$ the clinic estimates that the waiting cost in line for a children is $\backslash (\backslash$$ $20/\backslash )$ hour, what is the total waiting cost per hour?

vi$)$ the average number of children in the system if another health worker is added to the system $($assume only one waiting line in the system$)$ who has the same service rate as the existing one.

$($b$)$ At what rate must children be served in order to reduce the average time spent in the system to $\backslash (15\backslash$mathrm$\{$~min$\}.?\backslash )$