QUEUEING THEORY

WHAT IS A
QUEUE ?
A Queue is a linear structure which follows a particular
order in which the operations are performed.

QUEUEING THEORY
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Queuing theory is the mathematical study of queuing, or waiting in lines.
Queues contain customers (or “items”) such as people, objects, or information.
Queues form when there are limited resources for providing a service. For
example, if there are 5 cash registers in a grocery store, queues will form if more
than 5 customers wish to pay for their items at the same time.

Mathematical queuing models are often used in software and business to
determine the best way of using limited resources. Queueing models can answer
questions such as:What is the probability that a customer will wait 10 minutes in
line? What is the average waiting time per customer?

The following situations are examples of how queueing theory can be applied:
• Waiting in line at a bank or a store.
• Waiting for a customer service representative to answer a call after the call has been
placed on hold.
• Waiting for a train to come.
• Waiting for a computer to perform a task or respond.
• Waiting for an automated car wash to clean a line of cars

ASSUMPTIONS
1. Calling Population An infinite population with
independent arrivals and not
influenced by queueing system.
2. Arrival Process Poisson distribution of Arrival rate.
3. Queueing Configuration No. of waiting line with unlimited
space.
4. Queue discipline First come, First serve.
5. Service Process Exponential service of time
distribution.

KENDALL’S
NOTATION
• Single queueing nodes are usually
described using Kendall's notation in the
form A/S/c where A describes the
distribution of durations between each
arrival to the queue, S the distribution of
service times for jobs and c the number of
servers at the node.

SINGLE
SERVER
QUEUEING
MODEL
MULTIPLE
SERVER
QUEUEING
MODEL
TYPES OF QUEUEING MODEL
S I N G L E Q U E U E ,
S I N G L E S E RV E R
M U L T I P L E
Q U E U E , S I N G L E
S E RV E R
S I N G L E Q U E U E ,
M U L T I P L E
S E RV E R
M U L T I P L E
Q U E U E ,
M U L T I P L E
S E RV E R

The single queue with a single server and the single queue with multiple servers are
two of the most common types of queuing systems. However, there are two other
general categories of queuing systems: the single queue with single servers in
sequence and the single queue with multiple servers in sequence .

An example of a queuing system that has a single queue leading into a sequence of single
servers is the personnel office of a company where job applicants line up to apply for a specific
job. All the applicants wait in one area and are called alphabetically .The application process
consists of moving from one interview to the next in a single sequence to take tests, answer
questions, fill out forms, and so on. Another example of this type of system is an assembly line,
in which products are queued up prior to being worked on by a sequenced line of machines. If,
in the personnel office example, an extra sequence of interviews were added, the result would
be a queuing system with a single queue and multiple servers in sequence. Likewise, if
products were lined up in a single queue prior to being worked on by machines in any one of
three assembly lines, the result would be a sequence of multiple servers.

In a health clinic,The average rate of arrival of patients is 12 patients per
hour. On an average an doctor can save patients at rate of one patient
every 4 minutes.
 Assume that the arrival of Patients follows a Poisson distribution and
service to patients follow an exponential distribution.
 Find average number of patients in waiting line and in the clinic.
 Find average waiting time in waiting line or in queue and also the
average waiting time in clinic.

λ = Patients = 12/hr
µ = Doctor = 1 in 4 min = 15/hr
ρ = λ/µ = 12/15= 0.8
Ls = ρ/1- ρ = 0.8/1-0.8 = 4
Lq= ρ*Ls = 0.8*4= 3.2 = 3 approx.
Ws= Ls/ λ = 4/12 = 1/3 hr = 20 min.
Wq=Lq/ λ = 3.2/12= 0.266 hr = 16 min.

• We all have been to some supermarket or banks and we often face the problem of long
waiting time , that is always quite irritating.At times just to skip all this hustle we always
procrastinate the task .
• We all as customers want to wait for lesser time and the server definitely wants to make
maximum profit , here is where queuing theory comes in to provide to the analyst with a
powerful tool for designing and evaluating the performance of queuing systems.

PROBLEM
The present queuing system in the supermarket is 3 queues
and 3 servers, that is a single queue and single server
The manager wants to maximize the profits for the festive
season , hence wants a queuing system which much more
efficient .
Does there exist a better system such that the cost supermarket
bears is minimized and hence maximizing its profits ?

ASSUMPTIONS
Data for this study were collected from XYZ supermarket.
The methods employed during data collection were direct
observation and personal interview and questionnaire
administering by the researcher. Data were collected for 1
week.The following assumptions were made for queuing
system which is in accordance with the queue theory.
They are:
 Arrivals follow a Poisson probability
 The queue discipline is First-Come, First-Served
(FCFS).
 Service times are distributed exponentially.
 The queue limit is infinite.
 The service providers are working at their full capacity.
 The average arrival rate is greater than average service
rate.

 Expected total cost for M/ M/ 1 and M/ M/ S model: Service level is the function of two conflicting costs:
i) Cost of offering the service to the customers.
ii) Cost of delay in offering service to the customers.
 Economic analysis of these costs helps the management to make a trade-off between the increased costs of
providing better service and the decreased waiting time costs of customers derived from providing that service.

 Expected service cost E(SC)= sCs Where s is number of servers and Cs is service cost
of each.
 Expected waiting cost in the system E(WC)= LsCw Where Ls= Expected number of
customer in the system and Cw= cost of waiting by the customer.
 Expected total cost in the case multi queue-multi server model(i.e. S individual M/M/1
models) ,E(TC)= S(E(SC)+E(WC))=s(Cs+LsCw)
 Expected total cost in the case of single queue- multi server model (i.e. M/M/s model) ,
E(TC)= E(SC)+E(WC)=sCs+LsCw.
 The average number of customer in the bank: Ls= Lq +λ/µ

CONCLUSION
Thus we conclude that single queue
multi server is better in comparison
to multi queue multi server.The
waiting time of customers waiting in
the queue is reduced almost 3 times
to the previous one.We also proved
that expected total cost is less in
case of single queue multi server as
compared to multi queue multi
server model.

THANKYOU
PRESENTED BY-
MANSI ARORA (5020)
SRISHTI JAIN (5042)

QUEUEING THEORY

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