A simulation generated customer arrivals to a bank with clerks providing service. It provided arrival times $1$ and service times $1$ for customer $=1,2,...$

$($a$)$ Write an expression $($using summation notation $\backslash$Sigma $)$ for the average $($interarrival$)$ time, $,$ between arrivals based on simulation results. Show this simplifies to $(2*)/(1).$

$($b$)$ Let$$beaknownlongrunaverageservicetime$.$Ifthefirst$$servicetimesalternatebetween$$and$$

$(*=,\text{'}=,"=,0=,...,2=)$where$=$or$$if$$isoddoreven$.$Whatisthe

formula for sample variance $\text{'}$ given known $$ and what is its limit as $$ approaches infinity? $2$

$($c$)$ If service times are exponentially distributed, what should the simulation service time variance approach with increasing simulation time?