Case Study $2$: Queue Management at a Bank During Peak Lunch

Hours $(10$ marks$)$

A local bank experiences high customer traffic during the weekday lunch hour. To better

manage this surge, the bank has four tellers available to serve customers. Customers

arrive in a single line, and there is no limit on the queue length, allowing everyone to wait

until they are served. However, the bank is facing challenges with long waiting times and

needs to better understand its customer flow to optimize service and improve customer

satisfaction.

The bank has observed that during the lunch hour:

$$ An average of $120$ customers arrive at the bank.

$$ On average, there are $10$ people waiting in line at any given time.

$$ The tellers are $50\%$ utilized, meaning they are busy serving customers $50\%$ of the

time during this period.

The bank manager wants to determine how many people, on average, are in the system

during the lunch hour. This includes both those waiting in line and those being served

by the tellers. Understanding this will help the manager assess the efficiency of the

current system and make any necessary changes to improve service.

a$)$ Based on the information provided, calculate service rates of tellers. $[5$ marks$]$

b$)$ Using your answer in part $($a$),$ calculate the average number of people in the

system during the lunch hour. $[5$ marks$]$