Beate Klingenberg manages a Poughkeepsie, New York; movie theater complex called Cinema $8.$ Each of the eight auditoriums plays a different film; the schedule staggers starting times to avoid the large crowds that would occur if all eight movies started at the same time. The theater has a single ticket booth and a cashier who can maintain an average service rate of $280$ patrons per hour. Service times are assumed to follow a negative exponential distribution. Arrivals on a normally active day are Poisson distributed and average $200$ per hour.

To determine the efficiency of the current ticket operation, Beate wishes to examine several queue$-$operating characteristics.

a$)$ The average number of moviegoers waiting in line to purchase a ticket $=$ customers $($round your response to two decimal places$).$

b$)$ The percentage of time that the cashier is busy $=71$ percent $($round your response to the nearest whole number$).$

c$)$ The average time that a customer spends in the system $=\mathrm{.}75$ minutes $($round your response to two decimal places$).$

d$)$ The average time spent waiting in the line to get to the ticket window $=\mathrm{.}54$ minutes $($round your response to two decimal places$).$

e$)$ The probability that there are more than two people in the system $=0\mathrm{.}364($round your response to three decimal places$).$