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 $270$ 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.

Part $2$

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

Part $3$

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

Part $4$

$$c$)$ The average time that a customer spends in the system$=$ enter your response here minutes $($round your response to two decimal$$ places$).$

Part $5$

$$d$)$ The average time spent waiting in the line to get to the ticket window$=$ enter your response here minutes $($round your response to two decimal$$ places$).$

Part $6$

$$e$)$ The probability that there are more than two people in the system$=$ enter your response here $($round your response to three decimal$$ places$).$

Part $7$

The probability that there are more than three people in the system$=$ enter your response here $($round your response to three decimal$$ places$).$

Part $8$

The probability that there are more than four people in the system$=$ enter your response here $($round your response to three decimal$$ places$).$