Q $1-5.$ A dentist works from $9$ am to noon, offering $1-$hour appointments on the hour for routine cleanings $($the first patient arrives exactly at $9$ am$,$ the second patient arrives exactly at $10$ am$...).$ The dentist$$s schedule is always full, so she does not allow walk$-$ins. If she is free, the dentist begins the appointment when the patient arrives. The dentist finishes with each patient within $45$ minutes, and patients arrive at the beginning of their appointment.

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Question $12$ pts

How long on average do patients wait $($in minutes$)?$

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Question $22$ pts

The dentist hires an operations management specialist to analyze her appointment system. This analyst identifies that the time it takes the dentist to complete a cleaning is exponentially distributed with an average service time of $45$ min. How long, on average, do patients wait $($in minutes$)?$

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Question $31$ pts

Why is a patient$$s expected wait time different in the first versus the second scenario?

Group of answer choices

It isn't.

Because the dentist cannot fully balance the wait time caused by longer$-$than$-$average appointments with shorter$-$than$-$average service times.

Because variability in arrival times causes earlier$-$than$-$average arrivals to wait.

Because uncertainty in service times causes the dentist$$s utilization to increase.

None of the above

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Question $42$ pts

An operations management analyst has found that the time it takes the dentist to complete a cleaning has an exponential distribution with an average service time of $45$ min. The operations management analyst also considers the possibility of removing appointments and having walk$-$ins only. The analyst estimates the arrival process would be distributed exponentially with an average inter$-$arrival time of $60$ minutes. How long would patients expect to wait in this new system $($minutes$)?$

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Question $52$ pts

How many people should the receptionist expect to find in the waiting room in the scenario described in Q$4?$

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Question $61$ pts

Consider a stable queueing system with m identical servers. Which formula correctly describes the system utilization?

Group of answer choices

Arrival rate $$ service rate

Arrival rate $$ m $*$ service rate

Arrival rate$/($m$*$service rate$)$

Arrival rate$/$ service rate

None of the above