Respuesta :
Answer:
A) Single-server single-phase model (M/M/1).
[tex]\lambda=2.5 \,customers/hour\\\\\mu=6\,customers/hour[/tex]
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.
[tex]\lambda=2.5 \,customer/h[/tex]
The mean service rate is 6 customers per hour, calculated as:
[tex]\mu=\frac{60\, min/h}{10 \,min/customer}=6\, customer/h[/tex]
B) The average waiting time for a customer can be expressed as:
[tex]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[/tex]
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
[tex]\mu=\frac{60\, min/h}{8 \,min/customer}=7.5\, customer/h[/tex]
An the average waiting time for the service now becomes:
[tex]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[/tex]
The average time is now 4 minutes, so the goal is achieved.
