Respuesta :
Answer:
a) The average waiting time before the customer begins service is 13.5 minutes.
b) Number of customers are waiting to be served is 8.1.
Step-by-step explanation:
Given : Consider a queuing system with one server and infinite capacity. Suppose the arrival rate at the queue is 36 customers per hour, there are 9 customers on average in the system at any given time, and a server can serve 40 customers per hour.
To find :
a) What is the average waiting time before the customer begins service (in minutes)?
b) On average, how many customers are waiting to be served?
Solution :
A queueing system with one server and infinite capacity.
Let [tex]\lambda[/tex] be the arrival rate of customers i.e. [tex]\lambda=36/hr[/tex]
L be the average number of customers in the system i.e. L=9
[tex]\mu[/tex] be the number of customers a server can serve i.e. [tex]\mu=40/hr[/tex]
Average utilization of system is given by,
[tex]P=\frac{\lambda}{\mu}[/tex]
[tex]P=\frac{36}{40}[/tex]
[tex]P=\frac{9}{10}[/tex]
[tex]P=0.9[/tex]
Average time spent waiting in the system is given by,
[tex]W=\frac{1}{\mu-\lambda}[/tex]
[tex]W=\frac{1}{40-36}[/tex]
[tex]W=\frac{1}{4}[/tex]
[tex]W=0.25[/tex]
a) The average waiting time before the customer begins service is given by,
[tex]A_w=P\times W[/tex]
[tex]A_w=0.9\times 0.25[/tex]
[tex]A_w=0.225\ hr[/tex]
Converting into minutes,
1 hour = 60 minutes
0.225 hour = [tex]0.225\times 60[/tex] minute
0.225 hour = 13.5 minute
The average waiting time before the customer begins service is 13.5 minutes.
b) Number of customers are waiting to be served is given by,
[tex]n=P\times L[/tex]
[tex]n=0.9\times 9[/tex]
[tex]n=8.1[/tex]
Number of customers are waiting to be served is 8.1.
