1. A Project Report
on
SIMULATION BASED APPLICATION OF SINGLE SERVER
QUEUING PROBLEM
Submitted to
MANIPAL UNIVERSITY JAIPUR
BY
SHREY KAKKAR
(XII, NeerjaModi School)
For Summer Internship program
Under the Guidance of
Dr. Mohd. Rizwanullah
Asst. Professor-I
Department of Mathematics and Statistic, School of Basic Sciences
Manipal University, Jaipur, Rajasthan, India
2015

2. Table of Contents
Chapter No. Title
I Introduction:
1.1 Operations Research
1.2 Methodology or approaches to OR
1.3 OR models
1.4 Simulation: An Introduction
1.5 Steps of simulation process
1.6 Advantages and disadvantages of simulation
1.7 Types of Simulation and its Applications
1.8 Simulation Methods
II Problem under study
2.1 Statement of the problem
2.2 Scope and Objective of the study
III Research Methodology
3.1 Research design
3.2 Method and type of data collection
IV Data processing and analysis
V Findings
VI Recommendations

3. I. Introduction:
1.1 Operation research.
OR is the application of scientific methods, techniques and tools to problems involving the
operations of a system so as to provide those in control of the system with optimum solution
to the problem.
Much of this work is done using analytical and numerical techniques to develop and
manipulate mathematical and computer models of organizational systems composed of
people, machines, and procedures. It had its early roots in World War II and is flourishing in
business and industry with the aid of computer
Primary applications areas of Operations Research include forecasting, production
scheduling, inventory control, capital budgeting, and transportation, etc..
1.2 Methodology and approaches to OR.
The general steps/approaches for Operations Research problem solving are:

4. a) Definition
The first and the most important step in the OR approach of problem solving is to define the
problem. We need to ensure that the problem is identified properly because this problem
statement will indicate three major aspects:
i) A description of the goal or the objective of the study
ii) An identification of the decision alternative to the system
iii) The recognition of the limitations, restrictions and requirements of the system.
b) Construction
Based on the problem definition, you need to identify and select the most appropriate model
to represent the system. While selecting a model, you need to ensure that the model specifies
quantitative expressions for the objective and the constraints of the problem in terms of its
decision variables. A model gives a perspective picture of the whole problem and helps
tackling it in a well-organized manner. Therefore, if the resulting model fits into one of the
common mathematical models, you can obtain a convenient solution by using mathematical
techniques. If the mathematical relationships of the model are too complex to allow analytic
solutions, a simulation model may be more appropriate. There are various types of models
which we can construct under different conditions.
c) Solution
After deciding on an appropriate model we need to develop a solution for the model and
interpret the solution in the context of the given problem. A solution to a model implies
determination of a specific set of decision variables that would yield an optimum solution. An
optimum solution is one which maximizes or minimizes the performance of any measure in a
model subject to the conditions and constraints imposed on the model.
d) Validation
A model is a good representation of a system. However, the optimal solution must work
towards improving the system’s performance. We can test the validity of a model by
comparing its performance with some past data available from the actual system. If under
similar conditions of inputs, our model can reproduce the past performance of the system,
then we can be sure that our model is valid. However, we will still have no assurance that
future performance will continue to duplicate the past behaviour. Secondly, since the model

5. is based on careful examination of past data, the comparison should always reveal favorable
results. In some instances, this problem may be overcome by using data from trial runs of the
system. Note that such validation methods are not appropriate for non-existent systems, since
data will not be available for comparison.
e) Implementation
Our need to apply the optimal solution obtained from the model to the system and note the
improvement in the performance of the system. We need to validate this performance check
under changing conditions. To do so, we need to translate these results into detailed operating
instructions issued in an understandable form to the individuals who will administer and
operate the recommended system. The interaction between the operations research team and
the operating personnel reaches its peak in this phase.
1.3 Classification of OR models
OR models is broadly classified into the following types.
Physical Models: include all form of diagrams, graphs and charts. They are designed to
tackle specific problems. They bring out significant factors and interrelationships in pictorial
form to facilitate analysis. There are two types of physical models:
i) Iconic models
ii) Analog models
Iconic models are primarily images of objects or systems, represented on a smaller scale.
These models can simulate the actual performance of a product. Analog models are small
physical systems having characteristics similar to the objects they represent, such as toys.
Mathematical or Symbolic Models employ a set of mathematical symbols to represent the
decision variable of the system. The variables are related by mathematical systems. Some
examples of mathematical models are allocation, sequencing, and replacement models.
By nature of Environment: Models can be further classified as follows:
i) Deterministic model in which everything is defined and the results are certain, such
as an EOQ model.
ii) Probabilistic Models in which the input and output variables follow a defined
probability distribution, such as the Games Theory.
By the extent of Generality Models can be further classified as follows:
i). General Models are the models which we can apply in general to any problem. For
example: Linear programming.

6. ii) Specific Models on the other hand are models that we can apply only under
specific conditions. For example: we can use the sales response curve or equation as a
function of only in the marketing function.
1.4 Simulation
Simulation is a flexible methodology we can use to analyze the behaviour of a present or
proposed business activity, new product, manufacturing line or plant expansion, and so on
(analysts call this the 'system' under study). By performing simulations and analyzing the
results, we can gain an understanding of how a present system operates, and what would
happen if we changed it -- or we can estimate how a proposed new system would
behave. Often -- but not always -- a simulation deals with uncertainty, in the system itself, or
in the world around it.
1.5 Steps of simulation Process
No
Define the problem
Identify decision variables,
performance criterion and decision
rules.
Develop simulation model
Validate the model
Modify the model by
changing the input data, i.e.
Values of decision variables
Design experiments
Run or conduct the simulation
Is simulation process completed?
Examine the outputs and select the
best course of action
Yes

7. 1.6Advantages and Disadvantages of Simulation
GREAT FORECASTIN G POWER, BUT A GOOD THEORY IS NEEDED
Data analysis methods such as regression are limited to forecasting effects of events that
are similar to what has already happened in the past. For example, if a brand has been
investing in TV ads within a range of $50M to 29 $100M in the past few years, a
marketing mix model is excellent at forecasting what would happen if spend is within
those bounds. However, the model is likely to produce nonsensical results once it
extrapolates to forecast what would happen if TV spend is doubled or if a new marketing
channel is deployed.
Simulation has an advantage over these methods in that it allows us to forecast things that
have never happened before and to run scenarios outside of historical bounds. The caveat
is that we need a good theory and causal hypotheses about how the system we are
interested in analyzing works. Theories that have high predictive power, at least in social
science, are hard to come by and may take years to develop.
FLEXIBLE, BUT NOT STANDARDIZED
Simulations, and agent-based modelling in particular, provide highly flexible techniques
for answering a wide range of research questions. These questions include what happened
in the first moments of the Universe, how wind turbulence around aircraft works, how the
World Wide Web evolves, or how to better design hospitals. Although simulation can be
applied in a variety of contexts, a formalized set of rules and best practices is not always
readily available. For this reason, simulation modelling (especially in social science) is

8. incredibly creative, but may be daunting for new researchers who have no single
reference to consult when starting out.
1.7 Types of Simulation and its Applications
There are several types of simulation. A few of them are:
1. Deterministic verses time independent simulation: The deterministic simulation
involve cases in which a specific outcome is certain for a given set of inputs. Whereas
probabilistic simulation deals with cases that involve random variables and obviously
the outcome cannot be known with certainty for a given set of inputs.
2. Time dependant verses time independent simulation: in the former, it is not important
to know exactly when the event is likely to occur. For example, in an inventory
control situation, even if decision maker knows that the demand is three units per day,
but it is not necessary to know when demand is likely to occur during the day. On the
other hand, in time dependent simulation it is important to know the exact time when
the event is likely to occur.
3. Interactive simulation: This uses computer graphic displays to present the
consequences of change in the value of input variation in the model. The decisions are
implemented interactively while the simulation is running.
4. Corporate and financial simulations: this is used in corporate planning, especially the
financial aspects. The models integrate production, finance, marketing, and possibly
other functions, into one model either deterministic or probabilistic when risk analysis
is desired
Simulation is one of the most widely used quantitative methods -- because it is so flexible and
can yield so many useful results. Here's just a sample of the applications where simulation is
used:
 Choosing drilling projects for oil and natural gas
 Evaluating environmental impacts of a new highway or industrial plant
 Setting stock levels to meet fluctuating demand at retail stores
 Forecasting sales and production requirements for a new drug
 Planning aircraft sorties and ship movements in the military
 Planning for retirement, given expenses and investment performance
 Deciding on reservations and overbooking policies for an airline
 Selecting projects with uncertain payoffs in capital budgeting

9. 1.8 Monte Carlo Simulation
Monte Carlo simulation -- named after the city in Monaco famed for its casinos and games of
chance -- is a powerful method for studying the behaviour of a system, as expressed in a
mathematical model on a computer. As the name implies, Monte Carlo methods rely on
random sampling of values for uncertain variables that are "plugged into" the simulation
model and used to calculate outcomes of interest. With the aid of software, we can obtain
statistics and view charts and graphs of the results.
Monte Carlo simulation is especially helpful when there are several different sources of
uncertainty that interact to produce an outcome. For example, if we are dealing with
uncertain market demand, competitors' pricing, and variable production and raw materials
costs at the same time, it can be very difficult to estimate the impacts of these factors -- in
combination -- on Net Profit. Monte Carlo simulation can quickly analyze thousands of
'what-if' scenarios; often yielding surprising insights into what can go right, what can go
wrong, and what we can do about it.
II. Problem under Study
2.1 Statement of the Problem
A Dentist in Samaksh Dental Clinic, Raja Park, Jaipur works from Monday to Friday for 8
hours each, and schedules all his appointments for 30 minutes.
The following summary (Table-2.1) shows the various categories of work, their probabilities
and time actually needed to do the work.
Table – 2.1
Categories of services Time required
(minutes)
Probability
1 Scaling 45 0.18
2 Polishing 10 0.12
3 Filling 20 0.33
4 Extraction(non surgical) 30 0.07
5 Extraction(surgical) 60 0.03
6 Root canal 90 0.12
7 Braces 45 0.01
8 Dental implants 60 0.01
9 Dentures 60 0.01
10 Tooth replacement 90 0.12

10. Simulate the dentist’s clinic for eight hours and determine the average waiting time for
patients as well as the idleness of the doctor. Assume that all patients show up at the time of
scheduled arrival. Use the following random numbers for handling the problem-
39 76 45 90 64 26 31 58 97 06 99 73 23
88 87 71
2.2 Scope and objective of study
Since simulation techniques plays a very important role in solution of daily life problems, for
example effect of air turbulence in airplane, hospital management, computer systems and its
configurations, design of new shop floor etc., the problem under study gives an idea about the
average waiting time of the doctor (single server) and also helps to find the optimum number
of servers required.
The above problem can be generalized for any units based on single server which runs during
fixed number of hours.
III ResearchMethodology
3.1 ResearchDesign:
The problem under study is based on single server queuing model. Monte Carlo simulation
technique is applied with 16 random numbers for the study of 8 hours of dentist working in a
day.
3.2 Method and type of data collection
The data is collected in the form of primary data, interaction with doctor in the dental clinic.
Questionnaires has been designed to ask related to the problem under study
IV. Data Processing and Analysis
Under the Monte Carlo simulation technique, tag numbers/ random number intervals have
been designed under cumulative probability distribution for generating service time for each
patient from the dentist.

11. The Cumulative probability distribution Table:
Category of
service
Service time
required(min)
Probability
Cumulative
Frequency
Random number
interval
Scaling 45 0.18 0.18 0-17
Polishing 10 0.12 0.30 18-29
Filling 20 0.33 0.63 30-62
Extraction 1 30 0.07 0.70 63-69
Extraction 2 60 0.03 0.73 70-72
Root Canal 90 0.12 0.85 73-84
Braces 45 0.01 0.86 85-85
Dental implants 60 0.01 0.87 86-86
Dentures 60 0.01 0.88 87-88
Tooth
Replacement
90 0.12 1 89-99
Using 16 random numbers and above cumulative distribution table, we have generated
various parameters of a queuing system such as arrival pattern of customers’ service and
waiting time, in context of the given problem which is shown below.
Patient
number
Scheduled
arrival
Random number Category of service Time
required(min)
1 9.00 39 Filling 20
2 9.30 76 RCT 90
3 10.00 45 Filling 20
4 10.30 90 Tooth replacement 90
5 11.00 64 Extraction 1 30
6 11.30 26 Polishing 10
7 12.00 31 Filling 20
8 12.30 58 Filling 20
9 1.00 97 Tooth Replacement 90
10 1.30 06 Scaling 45
11 2.00 99 Tooth Replacement 90
12 2.30 73 RCT 90
13 3.00 23 Polish 10
14 3.30 88 Denture 60
15 4.00 87 Denture 60
16 4.30 71 Extraction 2 60

12. V. Findings
Average doctor waiting time- 10/16 = approx 1min/patient
Average patient waiting time= 2200/16 = approx 2hr 10min/patient
VI. Recommendations
 The doctor should keep a break between his working hours, i.e. from 2 to 5 pm
so as to reduce the waiting time of patients.
 The doctor should reduce his daily patients by 25%
 The doctor could also keep another assistant doctor for 3 hours (i.e. 5 pm to 8 pm
to increase efficiency and reduce patient waiting time.
Time Arrivals Departures Patient Number Waiting patients
9.00 1 1(20)
9.20 1
9.30 2 2(90)
10.00 3 2(60) 3
10.30 4 2(30) 3,4
11.00 5 2 3(20) 4,5
11.20 3 4(90) 5
11.30 6 4(80) 5,6
12.00 7 4(50) 5,6,7
12.30 8 4(20) 5,7,8
12.50 4 5(30) 6,7,8
1.00 9 5(20) 6,7,8,9
1.20 5 6(10) 7,8,9
1.30 10 6 7(20) 8,9,10
1.50 7 8(20) 9,10
2.00 11 8(10) 9,10,11
2.10 8 9(90) 10,11
2.30 12 9(70) 10,11,12
3.00 13 9(40) 10,11,12,13
3.30 14 9(10) 10,11,12,13,14
3.40 9 10(45) 11,12,13,14,15
4.00 15 10(35) 11,12,13,14,15
4.30 16 10(05) 11,12,13,14,15,16
4.35 10 11(90) 12,13,14,15,16
6.05 11 12(90) 13,14,15,16
7.30 12 13(10) 14,15,16
7.45 13 14(60) 15,16
8.45 14 15(60) 16
9.45 15 16(60)
10.45 16