The document discusses queuing theory and its application at Rao Dental Clinic. Queuing theory deals with analyzing systems where customers arrive for service and may need to wait in a queue. Observations were made of arrival and service times at the dental clinic. With one server, the utilization rate was too high, leading to long wait times. Adding more servers would reduce wait times by lowering the utilization rate. The document provides calculations to determine expected numbers of customers in the system and queue, as well as average wait times, under different service rates.

1. QUEUING THEORY
Applied on
Rao Dental Clinic

2. QUEUING THEORY
The study of the phenomena of standing,
waiting and serving is called Queuing Theory.
“Any system in which arrivals place demands
upon a finite capacity resource may be
termed as a queuing system.”- Kleinrock.
The queuing theory, also called the waiting
line theory is applicable to situations where
‘customers’ arrive at some ‘service station(s)’
for some service; wait; and then leave the
system after getting the service.

3. Applications of Queuing
Theory
Passage of customers through a
supermarket checkout,
Transfer of electronic messages,
Banking transactions,
Airport traffic,
Telecommunications,
Sale of theatre tickets.

4. General Structure of
Queuing System
Arrival
Process
Queue
Structure
Service
System
Customers
leave the
system

5. Elements of Queuing System
Arrival Process – Distribution that
determines how the task arrives in the
system.
Queue Structure – Order in which
customers are picked up from waiting line
for service.
Service System – (a) Structure of service
system.
(b) Speed of service.

6. Operating Characteristics of
Queuing System
Queue Length – The average number of
customers in queue waiting to get service.
System Length – The average number of
customers in the system.
Waiting Time in the Queue - The average
time that a customer has to wait in the queue
to get service.
Waiting Time in the System - The average
time a customer spends in the system from
entry to completion of service.

7. OBSERVATIONS
Arrival Rate
i. 2 minutes
ii. 1minute
iii. 2minutes
iv. 4minutes , two customers arrived simultaneously individually
v. 1 minute , two arrived in group
vi. 1 minute
vii. 2 minutes
viii. 5 minutes , two customers arrived
ix. 2 minutes
x. 1 minute
xi. 2 minutes
xii. 12 minutes
xiii. 5 minutes
xiv. 15 minutes
xv. 2 minutes ,arrived two
xvi. 3 minutes

8. Service Rate
• 8min took to serve completely
• 13 min
• 14 min took to serve completely
• 15 min to serve
• 10 min to serve
• After 2 min customer called
• After 3 min called
• After 8 min called
• After 5 min called
• After 20 min
• After 30 min
• After 40 min
• After 1 hour 15 min

9.  Around 15 customers were already waiting in the
queue.
 At a time 7 customers can be treated. So, it is a case
of MULTIPLE SERVERS.
 Customers were patient as it was a dental clinic and
it takes time to serve and treat customers.
 But there was one customer who wanted to be
served first, insisting that they just had to ask
something. So, were allowed to meet the doctor.
 Customers have to call the receptionist and take the
appointment, they are given a number and even the
approximate timings regarding when they should
come.
 This helps to reduce their waiting time and in a way
better manage the customers.

10.  Receptionist took the slips .
 Service is provided on First Come First Basis
according to the number given to customers when
they took the appointment..
 But service in RANDOM ORDER was also possible,
since as and when the dentists were getting free they
were calling the patients which are treated by them .
Thus, everyone had equally likely chance of being
called.
 Customers pass their time by watching the television
or playing /working on their mobile phones.
 Leaving the place completely is not possible , as it is
a dental clinic. So they cannot or hardly delay their
visit . Thus, ‘BALKING ‘ is not possible.

11. My Experience
It took 1 hour and 15 minutes for my turn
to come.
Although, I was served within 10 minutes.

12. Notations
λ : The arrival rate, which will have
units of arrivals per hour.
µ : The service rate, the variable 1/ µ
will have units of hours per customer,
so µ has units of customer per hour.
P = λ/µ : The traffic intensity (average
utilization) of the queuing system. So 1
– p = idle rate.

13. Queuing Models
(a) Probabilistic Queuing Models
• Poisson, Exponential, Single server infinite
model.
• Poisson, Exponential, Single server finite
model.
• Poisson, Exponential, Multiple server infinite
model.
(b) Deterministic Queuing Models
• Known regular interval arrival of customers.

14. Analysis
On the basis of observations,
Arrival Rate (λ) = 21 customers per hour
Service Rate (µ) = 6 customers per hour
Required,
1) The utilization parameter
2) The probability that the queuing system is idle?
3) What is the expected number of customers in the store?
4) What is the expected number of customers waiting for
shopping?
5) What is the average length of queues that have at least one
customer?
6) How much time should a customer expect to spend in the
queue?
7) What is the expected time a customer would spend in the
store?
8) Assuming that n>0 ( i.e. customers are in the system) what
is the probability that the waiting time (excluding service
time) of a customer in the queue shall be more than 10
minutes.

15. Calculation
At current rate ; λ > µ :
þ = λ / µ
þ = 21 / 6
þ = 3.5
This system is unviable ; Queue will
increase indefinitely.

16. If Server is Supplemented
with 3 Servers
λ = 21
µ = 24
þ = λ / µ
þ = 21 / 24
þ = 0.875
Now, the system becomes viable.

17. Average Number of People
in the System
Ls = þ/ 1 - þ
= 0.875 / 1-0.875
= 7
The number of patients are reduced to 7
in the clinic.

18. Average Number of People
in the Queue
Lq = þ² / 1 - þ
= (0.875) ² / 1- 0.875
= 6.125
= 6
The number of patients waiting for turn
are reduced to in queue.

19. Non Empty Queue
Lq’ = 1 / 1 - þ
= 1 / 1 - 0.875
= 8 minutes.
The system will be empty for 8 minutes in a
day.

20. Average Waiting Time in
Whole System
Ws = 1 / µ - λ
= 1/ 24 – 21
= 20 minutes.
The time required for patients in the clinic
is reduced to 20 minutes which could go
into hours earlier as experienced.

21. Average Waiting Time in the
Queue
Wq = λ / µ (µ - λ)
= 21/ 24 (24 – 21)
= 17.5 minutes.
The time required for patients for their
turn is reduced to 17.5 minutes which
could go into hours earlier as experienced.

22. Suggestions
The current system is unviable and can be
corrected only by increasing the number of
servers.

23. Prepared by:-
Anjali, Deepali,
Garima &
Gagandeep
M.Com (Pass)
Semester-2nd
Section-A
Group-B