$(3$ pts$)$ Arrivals to a gate for entry into a stadium are Poisson with an average interarrival time of $3\mathrm{.}2$ seconds. There are $6$ servers, each taking $17$ seconds on average to conduct a security check and scan a ticket. Service times are exponentially distributed.

$($a$)(1$ pt$)$ Assume each server has a separate line with arrivals equally distributed among servers. That is$,$ we have six $\frac{M}{M}\frac{?}{1}$ queues. For each server, what is the average utilization, the average cycle time for a fan, the average time in line for a fan, the average number of fans at each server $($including queue$)$ and the average queue length?

$($b$)(1$ pt$)$ Now assume a single line for all servers. For each server, what is the average utilization, the average cycle time for a fan, the average time in line for a fan, the average number of fans at each server $($including queue$)$ and the average queue length?

$($c$)(1$ pt$)$ Management is considering merging the arrivals to two identical gates. Each gate faced an arrival rate of one per $3\mathrm{.}2$ seconds as described in $($b$).$ The fan arrivals would double to the merged gate. What is the smallest number of servers that are needed at the merged gate?