The formula to calculate the expected average wait time for a M$/$M$/$c queue is:

Expected Average Wait Time $=$

Where:

ca $=$ Arrival rate per hour $=3$ customers$/$hour

cp $=$ Service rate per hour $=2$ customers$/$hour

u $=$ Utilization $=60\%=0\mathrm{.}6$

tp $=$ Average service time $=6\mathrm{.}8$ minutes $=0\mathrm{.}1133$ hours

c $=$ Number of servers $=2$

Explanation:

Arrival and Service Rates: The arrival rate $($ca$)$ is how many customers come per hour, and the service rate $($cp$)$ is how many customers each server can handle per hour. In your case, $3$ customers arrive per hour, and each server can serve $2$ customers per hour.

Utilization: Utilization $($u$)$ is the fraction of time a server is busy. Here, it's $60\%,$ meaning each server is busy $60\%$ of the time.

Average Service Time: This is how long it takes to serve a customer on average, converted into hours $($tp $=0\mathrm{.}1133$ hours$).$

Number of Servers: This is how many servers are working at the same time $($c $=2).$

Plugging in the values:

Expected Average Wait Time in hour $=$

Expected Average Wait Time $=14\mathrm{.}57$ minutes

Therefore, the expected average wait time for Spartan Redi$-$Care is approximately $15$ minutes. This is lower than the target wait time of $45$ minutes set by the facility manager.

Explanation:

Calculating Wait Time: We multiply the arrival rate by the average service time and then divide it by the difference between the number of servers and the utilization $($in decimal form$).$ In example, this results in about $0\mathrm{.}2428$ hours, or approximately $14\mathrm{.}57$ minutes.

Interpretation: This means, on average, a customer will wait around $14\mathrm{.}57$ minutes, which is well below the facility's target wait time of $45$ minutes.

So$,$ the calculation shows that the current setup is quite efficient in terms of customer waiting time.

The expected average wait time for customers at Spartan Redi$-$Care is calculated to be around $14\mathrm{.}57$ minutes, which is lower than the facility's target of $45$ minutes. This calculation is based on the number of customers arriving and being served per hour, the average service time, and the number of servers available.