Office Equipment Inc.: Waiting Line Model Analysis

This report analyzes the waiting line model, also known as queuing theory, to evaluate the business expansion strategy of Office Equipment Inc. The report examines the company's current business model, including technician service times, travel times, and customer arrival rates. Using queuing theory, the report calculates key metrics such as customer waiting times, the probability of no customers in the system, and the average number of customers in the waiting line and the system. The analysis provides recommendations for the company to hire additional technicians based on the expansion of their customer base. The report concludes that hiring an additional technician is beneficial for the company when expanding to 20 customers, and maintaining two technicians is optimal for 30 customers, resulting in cost savings. The report provides a comprehensive analysis of how the waiting line model can be applied to make informed business decisions and optimize service operations.

Running Head: WAITING LINE MODEL
WAITING LINE MODEL
Name of the student
Name of the University
Author’s Note

1WAITING LINE MODEL
Table of Contents
Introduction................................................................................................................................2
Discussion..................................................................................................................................2
Conclusion..................................................................................................................................5

2WAITING LINE MODEL
Introduction
Waiting line model also known as Queuing theory is a mathematical model which
studies the waiting line problems encountered by all the businesses nowadays. In simple
words, it deals with the issues arising because of the flow of customers. It is used in the cases
where wait time is to be minimized. This report aims to recommend Office Equipment Inc.
on its idea of business expansion in the coming two years with the help of the waiting line
mode or queuing theory.
Discussion
Current business Model of Office Equipment Inc. are as follows:
- It has only one technician to service 10 contracted customers
- Travel time is kept on an average of one hour
- Average service time is 1.5 hour with the cost of $80 per hour
- The company expects one call per 50 hours of operation
Arrival time of each customer (λ) = 1/ average rate of per call
= 1/50 = 0.02 calls per hour
Total service time= 1 hour (travel time) + 1.5 hour (repair time) = 2.5 hour
Average service rate or mean service rate per customer (μ) = 1/ total service time
= 1 / 2.5 = 0.4 customers per hour
According to the waiting line model, customers are expected to reach the service provider
hence, it is assumed that there is zero or no travel time.

3WAITING LINE MODEL
However, Office Equipment, Inc. states that their technicians will arrive to the
customers and hence, an average of one hour is kept to reach the customers. Although one
hour is considered as a part of service time, the customer has to wait for the first hour of his
service as the technician is traveling. Unlike the waiting line model that includes only service
time, in this case travel time will be added to the actual service time.
Probability that customers are in the system determines the time that the next
customer has to wait till his turn arrives. Lesser the probability of the existing customer, less
will be the waiting time of the next customer.
Probability that no customer is in the system = 1 - Probability that customers are in the
system
= 0.5380
Hence, approx. 50 per cent of the cases there will be no customer in the system.
Therefore, lesser waiting time, which is good for the company.
Average number of customers in the waiting line per hour and in the system are
0.2972 and 0.7593 respectively. Total number of customers are the summation of customers
in the waiting line and the number of customer that are being serviced in the system.
Average time the customer waits for the technician to arrive or in the waiting line =
1.6082 hours. Whereas average time till the time is back in operation is 4.1082 hours.

4WAITING LINE MODEL
Probability of waiting for more than one hour is 0.4620 and the total cost of service operation
per hour is $155.93.
One technician can meet 3 hours service call guarantee as the average time that a
customer waits is 1.6082 hours and average travel time taken by the technician is one hour,
which makes the total call waiting time as
 1 + 1.6082 = 2.6082
2.6082 is less than 3 hours hence, yes one technician can meet the service call within 3 hour.
Recommendations for 20 customers and 30 customers
For 20 customers For 30 customers
probability that no customers are in the
system
0.3525 0.1760
average number of customers in the waiting
line
0.2104 0.9353
average number of customers in the system 1.1527 2.3194
average time a customer waits until the
service technician arrives
0.5581 hours 1.6895 hours
average time a customer waits until the
machine is back in operation
3.0581 hours 4.1895 hours
probability that a customer will have to wait
more than one hour for the service
technician to arrive
0.2949 0.5600
The total cost per hour for service operation $275.27 $391.94

5WAITING LINE MODEL
If the company keeps on using one technicians even when the customer base is
expanded to 20 customers, the average waiting time per customer will increase to 6.9454
hours. The average travel hour taken by one technician is one hour, therefore the total waiting
time per customer go up to 7.9454 hours. Hours in a week when the technician is not on call
in case of one technician will rise to 42.3 hours (0.35250 * 1 technician * 40 hours) where as
if the company hire another technician it will come down to (0.35250 * 2 technician * 40
hours). Therefore, the company should hire another technician. Similarly, for 30 customers
with one technician number of hours in a week when the technician will not be on service
calls is 24.64 whereas with two technician it comes down to 14.08 hours.
In case of expansion of the customer base to 30 customers, the company should stay
with two technicians rather than hiring another technician as the cost of the company is
increased when it hires another technician and so, being with two technician reduces the
overall cost and waiting time of the customer. Annual saving of the company it stays with
two customer is $10,280 [($397.08 - $391.94) * 8 hours/day * 250 days/year].
Conclusion
Hence, from the above discussion it can be concluded that if the company decides to
expand its customer base to 20 customers then, it should hire one more technician to reduce
the total waiting time per customer and to reduce the overall cost of the company. Similarly it
was observed in case of 30 customers the company need not hire another technician as it will
increase the total cost of the company. Therefore, the company should stay with two
technician providing services to all the customers and have an annual saving of $10,280.