Answer:
A) Single-server single-phase model (M/M/1).
![\lambda=2.5 \,customers/hour\\\\\mu=6\,customers/hour](https://tex.z-dn.net/?f=%5Clambda%3D2.5%20%5C%2Ccustomers%2Fhour%5C%5C%5C%5C%5Cmu%3D6%5C%2Ccustomers%2Fhour)
B) The goal is not met, as the average time waiting for service is 5.56 minutes.
C) The new mean service rate is 7.5 customers/hour.
In this case, the average time waiting for service is 4 minutes, so the goal is met.
Explanation:
A) This situation can be modeled as a single-server single-phase model (M/M/1).
The mean arrival rate is 2.5 customers per hour.
![\lambda=2.5 \,customer/h](https://tex.z-dn.net/?f=%5Clambda%3D2.5%20%5C%2Ccustomer%2Fh)
The mean service rate is 6 customers per hour, calculated as:
![\mu=\frac{60\, min/h}{10 \,min/customer}=6\, customer/h](https://tex.z-dn.net/?f=%5Cmu%3D%5Cfrac%7B60%5C%2C%20min%2Fh%7D%7B10%20%5C%2Cmin%2Fcustomer%7D%3D6%5C%2C%20customer%2Fh)
B) The average waiting time for a customer can be expressed as:
![W_q=\frac{\lambda}{\mu}\frac{1}{\mu-\lambda} =\frac{2.5}{6}\frac{1}{6-2.5} =0.417*0.222=0.093\,hours\\\\W_q=0.093\,hours*(60min/h)=5.56 \,min](https://tex.z-dn.net/?f=W_q%3D%5Cfrac%7B%5Clambda%7D%7B%5Cmu%7D%5Cfrac%7B1%7D%7B%5Cmu-%5Clambda%7D%20%20%3D%5Cfrac%7B2.5%7D%7B6%7D%5Cfrac%7B1%7D%7B6-2.5%7D%20%3D0.417%2A0.222%3D0.093%5C%2Chours%5C%5C%5C%5CW_q%3D0.093%5C%2Chours%2A%2860min%2Fh%29%3D5.56%20%5C%2Cmin)
The average waiting time is 5.56 minutes, so it is more than the goal of 5 minutes.
C) If the average time spent per customer to 8 minutes, the mean service rate becomes
![\mu=\frac{60\, min/h}{8 \,min/customer}=7.5\, customer/h](https://tex.z-dn.net/?f=%5Cmu%3D%5Cfrac%7B60%5C%2C%20min%2Fh%7D%7B8%20%5C%2Cmin%2Fcustomer%7D%3D7.5%5C%2C%20customer%2Fh)
An the average waiting time for the service now becomes:
![W_q=\frac{\lambda}{\mu}\frac{1}{\mu-\lambda} =\frac{2.5}{7.5}\frac{1}{7.5-2.5} =0.333*0.2=0.067\,hours\\\\W_q=0.067\,hours*(60min/h)=4 \,min](https://tex.z-dn.net/?f=W_q%3D%5Cfrac%7B%5Clambda%7D%7B%5Cmu%7D%5Cfrac%7B1%7D%7B%5Cmu-%5Clambda%7D%20%20%3D%5Cfrac%7B2.5%7D%7B7.5%7D%5Cfrac%7B1%7D%7B7.5-2.5%7D%20%3D0.333%2A0.2%3D0.067%5C%2Chours%5C%5C%5C%5CW_q%3D0.067%5C%2Chours%2A%2860min%2Fh%29%3D4%20%5C%2Cmin)
The average time is now 4 minutes, so the goal is achieved.