# PNC Bank agency receives in average 20 customers per hour. Due to the current capacity, this agency has an average service time of 30 minutes every hour. The manager would like you to perform the following analysis. What is the probability of finding a busy system? Expected number of customers in the system Expected number of customers waiting in the system

PNC Bank agency receives in average 20 customers per hour. Due to the current capacity, this agency has an average service time of 30 minutes every hour. The manager would like you to perform the following analysis.
What is the probability of finding a busy system?
Expected number of customers in the system
Expected number of customers waiting in the system
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Solution:
Given, arrival rate, $\lambda =20$ per hour service rate $\mu =30$
Now,
1) Probability of busy system= $\frac{\lambda }{\mu }$also called $\rho =\frac{\lambda }{\mu }=\frac{20}{30}=0.67$
2) Expected no. of cestomers in the system= $\frac{\rho }{1-\rho }$
$=\frac{0.67}{1-0.67}$
$=2$
3) Expected no. customers waiting in the system
$=\rho ×\frac{\rho }{1-\rho }$
$=0.67×\frac{0.67}{1-0.67}$
$=1.34$