$2$ Points$)$ Using the Kendall notation scheme $($e$.$g$.$ M$/$M$/1),$ describe each of the

following queuing systems.

a$.(1$ Point$)$ Patients arrive at the Urgent Care center at a rate of $12$ per hour

$($Poisson distribution$).$ The treatment time follows an exponential distribution

with a mean of $30$ minutes, and the minimum number of providers is available to

treat patients.

b$.(1$ Point$)$ Patients arrive at the pharmacy randomly $($Poisson distribution$)$ at a

rate of six per hour to pick up prescriptions. Service time follows a general

distribution. There are two pharmacy techs on staff to help customers.

$4.(2$ Points$)$ Consider two systems with single server to serve patients. The following two

scenarios describe how the waiting time is affected by variability $($in patient$$ arrival time

and service time$)$ and utilization $($utilization is the demand rate over capacity rate$)$:

Scenario $1$: Suppose two systems have the same utilization rate. The system with

$\_\_\_\_\_\_\_\_\_($High$/$Low$)$ variability has shorter average waiting time.

Scenario $2$: Suppose two systems have the same variability. The system with $\_\_\_\_\_\_\_\_\_$

$($High$/$Low$)$ utilization has shorter average waiting time.

Hint: use the actual flow time $$ utilization graph

$5.(4$ Points$)$ Patients arrive at the walk$-$in clinic every $25$ minutes on average $($negative

exponential distribution$),$ and it takes about $20$ minutes for the physician to diagnose the

patient $($negative exponential distribution$).$

a$.(1$ Point$)$ On average, how many patients are there at the clinic?

b$.(1$ Point$)$ On average, how long do patients wait to see the physician?

c$.(2$ Points$)$ What is the probability that there are more than five patients at the

clinic?