### 1.1 Introduction to Modeling:

Introduction to Modelling: Modelling is an influential method l. With it, we can perform following functions.

• Analyze
• Design
• operate complex systems
• Hypotheses checking(Testing) with minimal cost (Performing the original events)

Modelling an effective communication Method/tool that tells us the happening of operation (how actually operations is done) and stimulates innovative rational about how to improve it.

These models shorten design cycles, minimize the expenses and increase the knowledge for industries like government, and educational institutions. We build the models to evaluate the real world problems which are complex to analyze through flowcharts and spreadsheets.

### 1.2 Queuing theory:

The mathematical study of waiting line is known as Queuing theory. In queuing theory, with the help of mathematical and statistical tools we analyze different processes. Thus we find different derivations and calculate the several performance measures which include,

• how much time spent in the system,
• what is the probability of bottleneck,
• how much customers are severed or waiting for service,
• Weather the system is completely occupied or partially or none of the customer is there waiting for service.

The calculations and results are utilized for business proposals, decision related to resources to complete certain tasks, that is why it falls under the domain of operations research.

It is used in different type of situation and industries like in commerce, business industry, public service and engineering. It has effective use in the transport and telecommunication.

### 1.3 Queuing Model:

In queuing theory, the real situations or system are shown in the form of queuing models which tells the behaviour of the original situation. The results are evaluated mathematically. We find following performance statistics by applying queuing theory.

• How much average number of customers in a line?
• How much average time used up in the lines?
• What is the algebraic pattern of these results?
• Weather the system is completely occupied or partially or none of the customer is there waiting for service?

The above statistical parameters are used to measure the customer satisfaction. These also help us in identifying the root causes of the queue problem and economic loss in business.

### 1.4 What is Simulation?

Simulation is defined as imitating a certain phenomenon (outward appearance or behaviour) by using another device. From the engineering point of view, knowing the actual behaviour of a system by making a duplicate having same characteristics as the original, with the use of a computer model is known as simulation. Simulation sometimes means experimentation with a scale model. In most cases a model is constructed within a computer and numerical experimentation is done by using the model. The device which executes simulation or the combination of computer and the internal software is known as a simulator. The main reason for the widespread use of simulation is the rapid and progressive development of the power of a computer. From the requirement point of view, a comprehensive understanding and evaluation of the subject systems has become essential in many fields because the systems used by society and industry are not only complex, but also massive, which complicates the analytical interpretation of the system as a whole and its experimentation. Simulation wherein a system is the object or simulation in system engineering is known as system simulation. This book focuses on system simulation. The utility of system simulation is not merely to interpret a problem, but in most cases to enable an in-depth understanding of it in order to find solutions by varying the inputs to the model and values of parameters. Such understanding is termed sensitivity analysis. In short, simulation is an indispensable measure for decision or judgment procedure and has become an important tool research and business. A simulation is not an objective per se but a tool or a procedure for attaining one. A problem is not solved simply through the results of simulation computations. The quality of simulation results cannot be determined solely by the simulation. Its utility or, in other words its validation, must be thoroughly assessed in view of the assigned objective and object. In the 1960s digital computers were put to practical use which brought about the development of several widely used simulation languages. These have gained ground in the world of large computers and are here to stay. The 1970s brought about significant utilization of simulators in various social and industrial fields and the development of simulators for exclusive use in various fields is continuing even now. In the 1980s, expansion in the development of simulators for exclusive use in various fields continued due to increased interest in the general purpose simulator, improvement in the performance of computers, and arrival of the supercomputer. On the other hand, several work stations and personal computers (PCs) appeared on the scene in the 1980s. Promotion of colour display and improvement in computer graphics technology stimulated visualization of computed results and usage of simulation techniques. User friendly simulation programs for work stations or PCs hit the market worldwide. One outcome of the availability of such programs was the ushering in of an era in which PC users and children can enjoy computer games which ingeniously make use of computer graphics and simulations. In the 1990s some people have become dissatisfied with the existing computers or simulation. Expectations from physicists for contributions to simulation are great. Computation physicists consider that by turning a computer into an experimental tool, the natural (physical) world and physical phenomena can be unravelled; they anticipate the arrival of a computer having a computing speed of 1 million times greater than the existing ones. The ideal of “computation experimentation is still a dream. However, if with progress computing speed can be made more rapid, and then real time animation with the supercomputer could probably be realized.

### 1.5 History of Simulation:

The analog type calculating machine was the first to appear in the history of computer simulation. Slide rules, planimeters, integrators, etc. have long been in use. The differential analyzer and the harmonic analyzer were created at the end of the nineteenth century. Integration was done by the product of rotational angle by means of a mechanical circular plate. In 1931, V. Bush invented the differential analyzer at MIT. It could also be called a mechanical type simple analog computer. Bush solved the highest 6th order differential equation with this [7]. Thereafter a device was developed which can automatically solve differential equations. Analog computers with vacuum tubes appeared on the scene around the time that World War II came to an end (1945). Who the inventor was is not quite clear but most people feel that this calculating device, having the same principle as the differential analyzer, can be achieved by a combination of vacuum tube circuits. In the 1950s, analog computers were investigated, studied and mostly used.

After the mid-1950s, when digital computers came into use, an idea was put forward to perform the simulation technique being carried out on an analog computer, on a digital computer as such. The software which embodied this concept was developed in various ways along with establishment of the general purpose large computer. .CSMP and DDS [8] were mostly used as the continuous system simulation languages. In the 1960s, simulation languages for the discrete system were also developed and GPSS and SIMSCRIPT are used even today. The beginning of the 1970s made use of the supercomputer possible and simulation of a natural phenomenon or a physical phenomenon advanced.

With the popularization of PCs and work stations during the latter half of the 1980s, in the development of several customized application programs, simulation programs, with excellent man-machine interface have been developed.

### 1.6 An Outline of Simulation:

When we talk of simulations, there are certain important items such as what is the objective of simulation, what steps are needed along with simulation as measures for accomplishing that objective, what are the entities of a simulation language used in a simulation model as well as simulation procedure and so on. These are shown in a Figure 1.1, which of course does not show all the items but selects the familiar heads of system simulation. Simulation is necessary at various points of time, such as time of system analysis (planning), time of designing, time of operation, after completion and so on, but mostly simulation is used at the time of planning and designing for pre-assessment of system in development, first of all qualitative assessment of the development subject and effect of development are carried out using such systems approach methods as scenario writing, brainstorming or the KJ (Kawakita Jiro) method, etc. For the essential parts of technology, performance is evaluated individually by experimentation. For actualization of simulation, there is need to pay attention to the many items shown in Figure 1.1. Depending on whether the variables of Simulation have to be handled with analogy, digital or hybrid (which combines both), it is decided whether to use an analogy computer, digital computer or hybrid computer. When constructing a model, there is need to properly understand the objective of simulation and to extract the distinctive features of the real world taken as the object. In the range of modelling one may either

1) Consider the actual world by dividing it into the object and the control scheme policy) to be applied to that object or

2) Imitate the object alone, excluding the control scheme or including it within the model? As far as the degree of approximation is concerned, there are fewer cases of using the primary model, which corresponds to the basic equation related to theory, but many cases of using the idea-related approximate model (secondary model). Actually occurring phenomena evolve with the passage of time and when the time variation is taken into account it becomes a dynamic model; when there is no need to take time variation into consideration, it becomes a static model. A dynamic model can be divided into continuous- change model and discrete-change model depending on the conception of time. The forward direction method which carries out simulation with the passage of time is the general method for simulation but sometimes reverse direction simulation is also done to seek the policy variable (corresponds to input in normal simulation) for realizing targets such as policy decisions. In both forward direction and reverse direction, repeated simulation is usually carried out by trial and error while changing the conditions. It is also possible to carry out optimization simulation by adding the function of optimization, which automatically seeks the optimum conditions. Depending on the aim of simulation, there are cases in which the time for proceeding simulation must be the same as the actual. For instance, driving training for aircrafts and vehicles must be proceeded in real time. Conversely, there are cases of shortening or extending the actual time. A large number of simulations are carried out with such an artificial time.

### 1.7 Objective of Simulation:

Simulation is used for understanding effects because use of an actual subject is not only costly but also involves certain risks. Besides, there is uncertainty regarding the time it will take to realize the results. The objectives of simulation include the following:

1) Qualitative understanding of the phenomenon occurring in the real world or the behaviour of a system is the first step to system engineering. If there is a model which can show the actual mechanism then new knowledge can be acquired by varying the inputs to the model or the parameters and doing repeated simulation. Utilization of simulation enables

1. Understanding the output response with respect to the input
2. Analysis of interfering conditions of two systems, e.g. the mechanical system and the electrical system
3. Sensitivity analysis of parameters or policy and so on.

These are shown in Figure 1.2 (a), (b) and (c). However, the model being used is not always necessarily correct. There is a method whereby several hypothetical models are given and experimentation is done with simulations and the correctness of the model is studied by verifying it against already known facts. This method is known as identification by simulation. The simulation used in physics research falls in this category. There are various methods for identification also, such as (d) identification of mere subject (e) cascade connection of identification system, (f) identification of cascade process and so on, as shown in Figure 1.2. Such types of simulations are quite often used for understanding and predicting not only natural phenomena, but also the existing state of social phenomena.

1) The system is developed after going through such processes as analyze, planning, designing and operation, But there is need to qualitatively analyze and understand the function and performance of the developed system beforehand in. each process. As an example of system development is the establishment of a chemical plant. Balance calculations with respect to incoming and outgoing material or heat balance are necessary for deciding the layout of the plant or configuration of the equipment. When installation of equipment is planned by simulation using such a static model, the quality of the operation control strategy can then be judged by operational simulation which makes use of a dynamic model.

2) Another objective of simulation is not just to examine the degree of satisfaction of function and performance, but also to evaluate the effect of system development on the system exterior by judging the quality of the overall system plan. This is known as environmental assessment or evaluation of negative impact.

3) There are also instances wherein human beings are actively included in simulation. In this case a system or model without human beings is well examined beforehand and already confirmed, after which humans are included in the simulation when the objective is to understand human behaviour and to provide educational training. Examining through experimentation the usability convenience for operating certain equipment used by people is an example of simulation for ascertaining human behaviour or psychological reaction. A flight simulator or business games are examples of educational training. Evacuation training at the time of prevention of building disaster constitutes not only educational training but also a means for understanding human behaviour under uncommon situations.

### Process Simulation:

Business Process simulation is the method that enables representation of processes, people, and technology in a dynamic computer model. In doing business process simulation, we follow four steps.

• Structure a model
• running a model
• evaluating the performance measures
• Assessing alternative scenarios.

The model mimics the operation of a business, by displaying the flow charts, animated pictures. The simulation software keeps the track of results and these results are used in evaluation process.

### 1.9 Discrete-System Simulation:

A real system is changing its state continuously. However, from the simulation point of view, when ,the change in state of the object system with the passage of time is considered as having occurred by events rather than continuously, it is termed a discrete system. During simulation, if the object system is modelled by considering it a system in which changes in state are taking place discretely, it is called discrete system simulation.

### 1.10 Simulation-based Optimization Techniques:

The main tool for designing the complex systems is discrete event simulation. Optimization techniques must be linked with simulation in order to design the system effectively. We show many optimization methods/ techniques. We find the doable practices for certain simulation model.

### 1.11 Simulation Software:

The new user is facilitated with the help of lots of new simulation software. Simulation software comprises the following items, like support, reactivity to bug notification, interface, etc.

A thing needed to keep in mind while we consider simulation software, like why do we need it, what should be the purpose, what should be the complexities, the foremost question should be why we need this? Does it for some industry or university or any sort of project for students?

After that we need to look in to some complex questions like what should be the main aspect of that program. Then we have to lay down some procedure to do so. What should be the input and output, what should environment. Do we need to train the people to handle certain tasks? How we can optimize the process. How we do the statistical calculation? Can we handle all these calculations? What is should be the cost of this process? It is very important to know which feature is the most important for the scenario.

### Table 1.1: Examples of simulation utilization in various fields

ELECTRIC POWER

Electric power system operation plan, water flow system operation plan, analysis of system characteristics, operation training

NUCLEAR POWER

System design, reactor core design, reactor core fuel design, operational control, physical phenomena (irradiation, plasma)

IRON AND STEEL

Plant layout plan, operation plan, management plan, stock management

PLANT

Facility plan, operation plan, scale model,

visualization of flow, vibrations, reliability and

safety analysis

TRANSPORTATION

City traffic plan, railway system plan, operational

control, plan for flow of objects on the premises,

simulated visual range

COMMUNICATION

Network configuration plan, transmission switching function, Network control function,

network management function

COMPUTER

Configuration plan, function (logic) evaluation, performance evaluation , operator/terminal evaluation

### SEMICONDUCTOR

Circuit design, device design, process design

Manufacturing

### MANAGEMENT

Support for Decision making, production management, stock management

### 1.12 Our Objective

The main purpose of this study is to provide a platform for modeling and simulation for the next coming student. Another underlying objective is to help and facilitate the industry to optimize their existing processes. Also helping them by the aid of demonstration of an optimization process we will build a model of a business process and then simulate it on appropriate software to analyze the existing position and its finest likely substitutions.

### 2.1 Definitions:

Modelling turns the real world into an abstract form and reproduces it as a concept which corresponds in form with the objective of simulation that which describes actually existing things and shapes or those parts of a system of particular interest, is known as a concept model or a thought model.

Embodiment of this concept model and turning it into a form which can be installed in a computer is called, in short, a computerized model.

While constructing the simulation model, due attention must be paid to integrating the idea of modelling and determining the range of the model. Modelling is a kind of process of conceptualization and since there are quite a few degrees of freedom in conceptualization, it is necessary to proceed with a consistent idea when modelling. Determining the range of the simulation model and selection of its structural elements are problems that can only be resolved when the properties of which part of which type of system to be investigated have been ascertained. The range and structural elements are related not only spatially but qualitatively as well. How the real world is to be apperceived at the time of modelling varies with the objective. There are two ways of comprehending the real world (Figure 2.1). One way sees the real world as comprising a really existing object and the policies controlling that object and the other looks at the real world as a whole, i.e., as a single object for analysis. The former is a modelling method used in simulation of the control system in design or in the process of policy planning. This approach is taken in the case of comparing and evaluating control policy alternatives by simulation. The latter aim at a status quo analysis for a state wherein separation between the object and. controls plan is insufficient, or when the objective is an in-depth understanding of the object for planning the control policy; in this case the modelling is cantered around the object and then simulated.

### 2.2 Modelling Procedure:

The modelling process for a large complex system such as the social system can be approximately separated into two steps: the first step is structural analysis to ascertain the type of structure in the object system or structural determination (identification); the second step quantifies the results and expresses them as a functional relation.

### 2.2.1 Structure Determination:

Structure determination (structuring) is the selection of primary factors (also known as elements or components) which need to be inducted into the model and analysis or identification of a connecting relation between these factors. For this purpose it is necessary to select the main factors from among several and to clarify the connecting relation among them. To include unrelated or remotely related factors would clutter up the model and result not only in a less accurate solution but demand more time in seeking it. The methods for selecting these factors include factor analysis, principle component analysis, multidimensional scaling analysis, etc. In addition, the KJ method, the ISM (interpretive structuring model), which falls under the graph theory and matrix method, the DEMATEL (decision making and trial evaluation laboratory) and the PPDS (planning procedure to develop systems) are well known [1].

### 2.2.2 Determination of Functional Relation:

After structuring has been established among the factors of the system, the functional relations to describe the relations among the variables of the model are determined. The functional relations comprise a static model which emphasizes the equilibrium state of the variables and expresses them by simultaneous equations, and a dynamic model using the differential or difference equation which lays emphasis on the temporal change in the variables. The economics model is a representative of the static model. Several engineering systems, in particular the control systems, broadly use the dynamic model.

Since a computer cannot make a selection regarding which are the main variables or which are the important functional relations, this necessarily depends on the accumulated experience and insight of the user. This is a vital prerequisite for doing effective simulation.

### 2.3 Simulation Models:

A simulation model usually shows how a system works? We use models to mimic the real life situations. With minimal cost we can test our hypotheses.

Say, we want to setup certain facility, we made a model of that and then run it to check the results. After that we can fine tune the model. Also we can add features at any time in the model.

### 2.4 Classification of Models:

There has not been much research regarding the science of classification of models or simulation (clustering of models). The reasons for this are several: the science of classification cannot produce a realistic clustering, the objects of simulation are diverse, and hence it is difficult to obtain a technical uniformity among them. A classification of a model in terms of degree of approximation, degree of abstraction, temporal representation, model characteristics and so on is given below [2, 3].

### 1) Classification According to Degree of Approximation:

Ø Micro model

Ø Macro model

Ø Primary model

Ø Secondary model

### 2.4.1.1 Primary Model:

A primary model means a model without approximation such as fundamental equations. The primary model encompasses the most fundamental theoretical laws or axioms and corresponds to the material point receiving the application of the three axioms of Newtonian mechanics, universal gravitation and the DNA model in biology.

### 2.4.2.1 Secondary Model:

The secondary model aims to act as a mediator between the real world and the theoretical world and makes the application of theory easy by selecting the elements which are important in reality and omitting the less contributory elements; it constitutes one of the approximation ideology related models.

In the case of simulation of a large-scale system, the secondary model is mostly used due to easy comprehension and reduced calculation time. The creation of a suitable model is linked with achieving successful simulation. A secondary model being an approximation model, complimentary and antagonistic secondary models exist together.

### 2) Classification According to Degree of Abstraction of the Model:

Ø Physical Model

Ø Actual model

Ø Scale model

Ø Analogy model

Ø Analog model.

Ø Mathematical Model

Ø Digital model

Ø Statistical model

Ø Analytical model.

Ø Logic model

Ø Structural model

### 3) Classification According to Temporal Representation:

Ø Static model

Ø Dynamic model

Ø Continuous-change model

Ø Discrete-change model

Ø Combined Model

### 2.4.3.1 Static models:

These are represented in the form of equations. In that we find the effect of each substitute by one time computation of equation. We take variable as averages. By adding the individual effects, we find the results. For example, Spread sheets are static models.

### 2.4.3.2 Dynamic models:

Simulation is known as dynamic modelling. It is represented by software, which do multiple calculations as opposed to static model in which only one calculation is done on equation. It is time based model, so whenever the time changes, it recalculates the model.

### 2.4.3.3 Continuous-Change Model:

The method for building a model by the continuous-change model perceives the subject system to be changing continuously and depicts the change in the system at tiny equal intervals of time.

A fluid flowing through the pipe is good example of Continuous simulations. In continuous flow volume may vary i.e. increase or decrease, with the passage of time values changes.

The simulation languages used for the representation of continuous change model are CSMP and DYNAMO.

### 2.4.3.4 Discrete-Change Model:

The method for building the model by the discrete-change model pays attention to any significant event occurring within the system and portrays the system by connecting events together. The simulation time is forwarded at unequal intervals of time at each occurrence of the event.

We consider the patient flow in a clinic. In that the clinic could be empty or have any number of patients moving through it.

The patients come out in random intervals. On the other exit of clinic, Actions happening and inside the clinic, would finds what comes out and when.

During the simulation discrete entities change state as events occur. Patient arriving, passengers arriving and in banks customers service calling are few examples of discrete events.

### 4) Classification According to Model Characteristics:

Ø Linear model

Ø Nonlinear model

Ø Deterministic model

Ø Stochastic model (Monte Carlo method, gaming method)

### 2.4.4.1 Deterministic Model:

A simulation model is properly used depending on the circumstances of the actual world taken as the subject of consideration. A deterministic model is used in that situation wherein the result is established straightforwardly from a series of conditions. In a situation wherein the cause-and-effect relationship is stochastically or randomly determined, the stochastic model is used.

A deterministic model has no stochastic elements and the entire input and output relation of the model is conclusively determined. A dynamic model and a static model are included in the deterministic model.

Simulation by the deterministic model can be considered one of the specific instances of simulation by the stochastic model. In other words, since there are no random elements in the deterministic model, simulation can well be done just once. However, in case the initial conditions or the boundary conditions are to be varied, simulation has to be repeated by changing the data. On the other hand, in Monte Carlo simulation, once the value has been decided by extracting a random number, the simulation does not differ from deterministic simulation.

### 2.4.4.2 Stochastic Model:

A stochastic model has one or more stochastic elements. The system having stochastic elements is generally not solved analytically and, moreover, there are several cases for which it is difficult to build an intuitive perspective. In the case of simulating a stochastic model, a random number is normally generated by some method or the other to execute trial. Such a simulation is called the Monte Carlo method or Monte Carlo simulation.

In case the stochastic elements in the simulation are two or more persons and there is a competitive situation or some type of game being reproduced, this is specifically known as gaming simulation.

### 5) Classification According to Application of the Model:

Ø Company model

Ø Economics model

Ø City model

Ø Industrial model

Ø Environment model

Ø Production model

Ø Plant model

Ø Engineering model

Ø Mechanical model and so on.

### 2.6 Model evaluation:

A very important step in modelling process is to measure the usefulness of a model, how it works, does it shows the properties of a system for which we have developed the model, does the model shows all events which are measured and which couldn't be measured (Extrapolation).

Usually we divide data in to two data sets.

• Training data

• Authentication data.

We use the training data to train the model. While to evaluate model performance we use verification data. Let us assume that the training data and verification data are not similar. Now if the verification data is best fit on the model, then the model defines the real system well.

### 3.1 Queuing:

Queuing was initially used by Agner Krarup Erlang, An engineer from Denmark, who was employed in the Copenhagen Telephone Exchange, published the first paper on queuing theory in 1909. In 1953, queuing code was introduced by David G. Kendall.

### 3.2 Elements of Queuing Systems:

Following are the queuing elements.

### 3.2.1 Population of Customers:

Population may be limited (closed system) or unlimited (open system). An imaginary model with big numeral of likely customers is considered to be unlimited population e.g. a hospital; on busy road, a motorway toll plaza. While in the scenario of a limited population, it may be limited to certain amount or quantity.

### 3.2.2 Arrival:

The way of customer's entry in to system is defined by Arrival. Usually the random arrivals have random intervals among two contiguous arrivals. The arrival pattern is shown by random distribution.

### 3.2.3 Queue:

How many customers are waiting for service is described as Queues. Normally the customers which are attended are not assumed to be the part of the line. There are two essential features of a queue: Maximum Size and Queuing Discipline.

### Maximum Queue Size (also called System capacity):

The number of maximum client in a queue depicts the system capacity; it may be limited or unlimited. Mostly its limited in nature, while in theory, we may assume the unlimited system capacity. The limited capacity of system may cause rejected without being served.

### Queuing Discipline:

How a queue gets a shape comes under the domain of queuing discipline. (Guidelines of injecting and eliminating customers to/from the queue). Following are the ways to represent the queue.

1. FIFO (First in First Out) also called FCFS (First Come First Serve) - orderly queue.

2. LIFO (Last in First Out) also called LCFS (Last Come First Serve) - stack.

3. SIRO (Serve in Random Order).

4. Priority Queue that may be observed as a number of queues for several priorities.

### 3.2.4 Service:

An activity takes some time to complete, that time is called service time, while other customer waits during that time. It should be taken as general, as it might be a client or machine. Service pattern are described as the theoretical models which are based on random scattering of service interval. An additional significant factor is the number of servers. These are of two types, single channel system (Systems with one server only), and multi-channel systems (systems with more servers).

### 3.3 Notation:

In 1953, David G. Kendall introduced the symbolization for labeling the features of a queuing model. He introduced in the form of an

### A/B/C

queuing code that can be seen in all standard modern works on queuing theory, for example, this code shows A as inter-arrival time distribution, B as service time distribution and C as number of servers.

In actual a queue is written in shorthand code by

### A/B/C/K/N/D

or the briefer

### A/B/C

. In this briefer version, it is assumed K = ∞ and N = ∞ and D = FIFO.

### 3.3.1 Kendall's Coding:

In most reference work about queuing theory, these types of symbols appear.

### 3.3.1.1 The Arrival Process: (A)

Arrival process is described by A. The codes used are:

### Description

M

Markovian

Poisson process (or random) arrival process.

MX

Batch Markov

Poisson process with a random variable X for the number of arrivals at one time.

MAP

Markovian arrival process

Generalization of the Poisson process.

BMAP

Batch Markovian Arrival Process

Generalization of the MAP with multiple arrivals

MMAP

Markov modulated poison process

Poisson process where arrivals are in "clusters".

D

Degenerate distribution

A deterministic or fixed arrival time.

Ek

Erlang distribution

An Erlang distribution with k as the shape parameter.

G

General distribution

Although G usually refers to independent arrivals, some authors prefer to use GI to be explicit.

PH

Phase-Type Distribution

Some of the above distributions are special cases of the phase-type, often used in place of a general distribution

### 3.3.1.2 The Service Time Distribution: (B)

The service time distribution is described by B. A number of common notations are:

### M

Markovian

Exponential service time.

### D

Degenerate distribution

A deterministic or fixed service time.

### Ek

Erlang distribution

An Erlang distribution with k as the shape parameter.

### G

General distribution

Although G usually refers to independent arrivals, some authors prefer to use GI to be explicit.

### PH

Phase-Type Distribution

Some of the above distributions are special cases of the phase-type, often used in place of a general distribution

### 3.3.1.3 The Number of Servers: (C)

The number of service channels or servers is represented by C.

### 3.3.1.4 The Number of Places in the System: (K)

The letter K shows about the capacity of the system or the maximum number of customers tolerable in the system containing those in service. The further addition of client is rejected when the number is at this maximum. The capacity is expected to be unlimited, or infinite if we omit this number.

Note: This is occasionally symbolized C+k where k is the buffer size, the number of places in the queue above the number of servers C.

### 3.3.1.5 The Calling Population: (N)

It is the overall population from which customer come to get the service. The population may be assumed to be unlimited if we assume this number.

### 3.31.6 The Queue's Discipline: (D)

This symbol describes about the Service Discipline or Priority orders that jobs in the queue, or waiting line, are attended:

### FIFO/FCFS

First In First Out/First Come First Served

The customers are served in the order they arrived in.

### LIFO/LCFS

Last in First Out/Last Come First Served

The customers are served in the reverse order to the order they arrived in.

### SIRO

Service In Random Order

The customers are served in a random order with no regard to arrival order.

### PNPN

Priority service

Priority service, including preemptive and non- preemptive.

### PS

Processor Sharing

## Kendall's Notation for Queuing Models:

### Queuing models

can be represented using Kendall's notation:

A/B/S/K/N/Disc

Where:

• A is the inter-arrival time distribution

• B is the service time distribution

• S is the number of servers

• K is the system capacity

• N is the calling population

• Disc is the service discipline anticipated

Mostly the last notation members are removed, so the notation get the shape of A/B/S and it is assumed that K = ∞, N = ∞ and Disc = FIFO.

A few basic notations for distributions (A or B) are:

• M for a Markovian (exponential) distribution

• Eκ for an Erlang distribution with κ phases

• D for Degenerate (or Deterministic) distribution (constant)

• G for General distribution (arbitrary)

• H for a Phase-type distribution

### 4.1 Simulation:

Simulation is a commanding tool for following three functions.

• Examining

• Planning

• Operation of complicated systems

It is almost costless method to test our thoughts and hypotheses. With this we can save hundreds and thousands of dollars.

The real life is show in form of simulation which mimics the real life situation. In simulation we show the general characteristic of real life situation into simulated model.

We use simulation in numerous situations, containing the modelling of natural systems or human systems in order to gain insight into their functioning. With the help of simulation we can test alternatives and different course of actions. .

### 4.2 History of Simulation:

The analogy type calculating machine was the first to appear in the history of computer simulation. Slide rules, planimeters, integrators, etc. have long been in use. At the end of 19th century, the differential analyzer and the harmonic analyzer were created. In 1931, V. Bush invented the differential analyzer at MIT. It could also be called a mechanical type simple analogy computer. Bush solved the highest 6th order differential equation with this. Later a device was made which can automatically solve differential equations. Analog computers with vacuum tubes appeared on the scene around the time that World War II came to an end (1945). Who the inventor was is not quite clear but most people feel that this calculating device, having the same principle as the differential analyzer, can be achieved by a combination of vacuum tube circuits. In the 1950s, analogy computers were investigated, studied and mostly used. After the mid-1950s, when digital computers came into use, an idea was put ahead to perform the simulation technique being carried out on an analogy computer, on a digital computer as such. The software which embodied this concept was developed in various ways along with establishment of the general purpose large computer. CSMP and DDS were mostly used as the continuous system simulation languages. In the 1960s, simulation languages for the discrete system were also developed and GPSS and SIMSCRIPT are used even today.

### 4.3 Types of Simulation:

Following are different types of simulation.

### 4.3.1 Physical Simulation:

In this type of simulation we use real physical things to perform different actions. The physical object is much cheaper than the real one.

### 4.3.1.1 Interactive Simulation:

The best example of this type of simulation is Flight simulator in which human is involved to perform and verify the results.

### 4.3.2 Computer Simulation:

In that we use computer to model the real life situation. With the help of computer we can see how a model works. There is ease of changing variables to get different results.

Now a days modelling of any natural systems in any discipline of science and business education as well as in engineering has become very important and is done through Computer simulation. This will tell us the inner picture of any process for our better understanding.

The mathematical model gives analytical solution which enables us to forecast the behaviour of the system with fixed set of conditions. But with the advancement of computers, it substituted the old fashioned model with fixed inputs. There is different type of simulations.

Monte Carlo simulation and stochastic modelling like software which are computed based programs enables us to model the process and it is very easy to do so.

### Figure 1:

Three Sub-Fields of Computer Simulation.

In order to simulate something physical, we build a mathematical model which is a representation of real object. Models can get the shape of declarative, functional, constraint, spatial or multimodal.

The next step is to run the model on computer. For that we have to develop a computer program. After execution we find certain results and data, which are then analysed. This is an iterative process and repeats itself and refines the result by using different alternatives.

### 4.4 Simulation and Analytical Method:

Simulation methods can be broadly classified into three types.

### 1.

The first type uses the actual system itself and carries out simulation by using a scale model and is called the direct analogy. As an extreme example of the first type, the direct analogy, an actual system can be used as the model. Only the phenomenon occurring in the real system is replaced by a replica or an imitation to study the system characteristics. For instance, carrying out disaster prevention training by reproducing the occurrence of a disaster is applicable to this type. In the direct analogy, scale models are mostly used, such as wind tunnel experimentation for aircrafts to examine the hydro-chemical characteristics, water tank experimentation for ships, and experimentation for positioning the equipment or ducts within a nuclear reactor chamber and so on. Broadly speaking, experimentation with animals in medicine could also come under the direct analogy. However, in system simulation, the direct analogy is by and large excluded.

### 2.

The second type simulates a version of the real world simplified into some model. The second type of simulation, using a mathematical model, employs a model made from the real world and numerical value is actually applied to that to reorganize the real. Most cases of system simulation fall in this type. This type of simulation is not restricted to engineering cases only, which handle physical variables; it is also used in the fields of social sciences, management, etc. In a large-scale system or in the social sciences it is difficult to experiment with the real system and experimental data are mostly generated by simulation. In other words, by applying different variables to the model and doing simulation, it is possible to perform numerical experiments easily in response to different conditions.

### 3.

The third type is a situation wherein the problem is formulated in an analytically solvable form by further abstraction and solved numerically by simulation. The broad relationship among these three types of methods and the analytical method is shown in Figure 1.3 [1].

The third type of simulation, using an analytical model, does not analytically solve the mathematical formula but obtains numerical results by intentionally using simulation. This Enables simulating a stochastic model by the Monte Carlo method; numerical solving of the probability phenomenon in the real world becomes possible by replacing even the originally non- probability type of partial differential equation or integral equation by a stochastic model. However, since this method is not always precise nor as manageable compared to other numerical analysis methods, its usability is not always high.

### 4.5 How is simulation performed?

The simulation is mostly performed on computers by writing computer program. Simulation can be performed manually.

With the advancement of technology, companies are in race to produce high performance simulation softwares for the industry. This made companies to announce their products on weekly bases. Such rapid growth not only produced certain advantages but also produced some disadvantages. These are as follows.

Simulation is more than just look into the future. There are certain advantages mention by different authors (Banks, Carson, Nelson, and Nicol (2000); Law and Kelton (2000); and Schriber (1991)) and include the following:

• With the help of Simulation, we can choose the correct option without utilizing extra resources. Without utilizing any sort of resources Simulation allows us to test our designs.

• We can hold the length of time by compressing or expending. We can utilize less or more time to get the desired results.

• With the help of simulation we can get knowledge of a system, how it works.

• Simulation enables us to use different alternatives and choose the best one.

• Simulation helps us in finding different problem in our model.

• Simulation helps the industry to remove different bottleneck.

• Simulation enables us to understand the process and visualize the plan. We can develop different animations and drawings to get better understanding of job.

• With the advancement of technology, we are now performing Simulation much faster than the past.

• With the help of simulation we can train people for real life situations.

Following are few disadvantages associated with the simulation, are:

• To create a prototype, it needs lot of preparation and experience.

• An output observation is usually random but it is difficult to find out that weather the system is random or not.

• If we just consider the cost of the modelling and cost to analyze the data, it might result in poor estimation; with this approach we cannot get the optimized results.

• Another disadvantage is that we can only use simulation in that case where it is possible to have analytical solution. A small queuing system and some probabilistic inventory systems, for which closed-form models (equations) are available are good examples of this.

• Although closed-form models are useful for small queuing and inventory problems, most real-world problems are too complex to be solved with these approaches. Simulation is necessary when there are a large number of events and interactions in a system, which is true of most manufacturing problems.

Introduction of the Organization Shaukat Khanum Memorial trust is a charitable organization. The Trust has set up a state of the art cancer hospital at a cost of US \$22.2 million on a 20 acre. Land located at Johor Town, Lahore Pakistan. The prime objective of the hospital is to provide free treatment to deserving cancer patients. The hospital is the only institution in the country providing diagnostic and treatment facilities to cancer patients under one roof. The hospital employees over 550 persons dedicated to provide highest quality services of international standards. Continuous quality improvement programs are an integral part of the hospital function. The Doctors, nurses and technicians have been employed from overseas. These include expatriates from USA, UK, Australia and Philippines.

The hospital is affiliated with Christie Hospital NHS Trust, Manchester, UK and University of Kentucky Medical Centre, Lexington, Kentucky, USA to share knowledge, skills, expertise and experience with the aim of improving and enriching the lives of citizens.

Pakistan, a Country of 130 million people generates over 700,000 new cases of cancer annually. Many people do not have access to even the most elementary healthcare. As a result, cancer which could be treatable in the West is often not diagnosed until it's too late. Even when they are diagnosed, treatment is too expensive. The average family in Pakistan finds even the most basic medical treatment a severe drain on their financial resources. For these people, paying for cancer treatment is out of question. So, a fundamental principle of the Shaukat Khanum Memorial Trust mission is to provide free treatment to poor cancer patients.

This thesis was initiated with the aim of examining the current operational aspects of Shaukat Khanum Memorial trust (SKMT) and then applying queuing theory, so that we can make feasible recommendations to enhance the operational efficiency and improve patient service level.

### Objectives:

The objective of study can be enumerated as follows:

1. To conduct a study on the operational aspects of some business organization (Profitable or non-profitable).

2. To generate some feasible solution to the operational problems faced by Shaukat Khanum Memorial trust and enhance the effectiveness and efficiency.

### Methodology

The study approach comprised main of interviews with staff of Hospital and one week field study cum primary data collection at Hospital. This time period was representative of normal working conditions.

Patient arrival rates, service rates and waiting times at each service station were collected using contrived observation method in accordance to instructions. The hospital arrival figures were collected from the operations department.

The arrival patterns of the hospital attendance are analysed. Capacity requirements analysis and queuing theory analysis are performed. The capacity utilization and expected waiting time at each station are presented and discussed.

### SKMCH & RC QUEUING CHARACTERISTICS

In this case, the process is complex. There is single queue. System has multiple phases. In some phases it has multiple servers' i.e. These servers provide same service with more than one counter. Queuing discipline is FCFS.

### No of servers

Information and Reception Counter

1

OPD Counter

2

Cashier / Finance Counter

2

Clinic

10

### OPERATION

SKMCH & RC OPD (out-patient department) operates as follows. SKM OPD has divided appointment system in two phases.

### REGISTRATION PHASE:

In that phase when patient enters into the system he is directed towards main OPD reception counter by Patient guides. As he approaches the information and reception counter, he is welcomed by the PCO (Patient Care Officer). At that counter he is given information regarding the procedure and directed to registration counter. Registration counter carries three operations.

1. Generating data base of patient

2. Generation Work order

3. Management of appointment

### Diagram -1

The new patient has to go through the whole process as shown in the diagram-1. On arrival first go to reception counter then to registration desk and generate work order and then for payments, goes to Finance counter while the follow up patient goes straight to Appointment counter and follows the other path shown in diagram -2.

### APPOINTMENT PHASE:

SKMCH & RC OPD opens from Mondays to Fridays (Five days a week). OPD opens at 9 a.m. and closes at 5:00 p.m. daily while it has lunch break of 1 hour from 1 pm to 2 pm. Therefore working hours are 7.

When a patient is arrived at the OPD and approaches the information counter he is guided to OPD counter. At that counter, PCO (Patient care officer) guides for appointment and sends patient file to nursing counter. There are two nursing counter, which manage the patients file and patient has no direct interaction with this counter.

### Diagram -2

Now the patient is sitting in waiting area for his turn. Upon call from nursing counter in chronological order, the nurse from nursing counter guides the patient to clinic

In clinic nurse takes necessary vitals like blood pressure, temperature, weight and heights etc. Then Medical Officer takes the history and then finally consultant sees the patient. After getting treatment patient may be sent to either diagnostic center for tests and to pharmacy or released to go home. If patient is sent to diagnostic center, they have to first go to finance department for the payment of test fees. Then they are headed towards the diagnostic center and pharmacy too.

In the Next section, the data is collected on patient's arrival time and doctors' service times are analysed.

### DATACOLLECTION

Data analysis provides the driving force for any simulation model. In this study, there types of data were collected: (This data is taken from Appendix)

 1st Week 2nd Week 3rd Week 4th Week Average(Week) Monday 403 416 337 331 372 Tuesday 359 414 359 405 384 Wednesday 367 347 364 379 364 Thursday 402 404 298 411 379 Friday 204 211 232 226 218 Total 1735 1792 1590 1752 343 8586

### Table 2

(Total number of patient per and average number of patients per day)

Information

35

30

117%

OPD

30

8

375 %

Cashier

8

60

13 %

Clinic

8

16

50%

i)

### Patients' arrivaltimes:

Data collected from observation and the hospital record over a period of four weeks. Keeping in view the difficulty related to recording arrival of patients, the arrival time for each patient was set at the time the patient arrived at the Information counter. The arrival rate of weekdays and weekends is calculated from the data provided in appendix. This gives hourly rate, daily rate on each day, weather it is weekend or weekday. Average daily arrivals are 343 patients. Total weekly arrivals were 8586 patients. Patient arrival distribution is passion while inter-arrival distribution is beta. We find arrival time as 35 patients per hour.

9 am-10 am

40

10 am-11 am

50

11 am-12 am

50

12 am -1 am

60

1 am -2 am

Lunch Break

2 am-3 am

55

3 am-4 am

50

4 am -5 am

38

### Table 5

(Average number patients per hour)

ii)

### OPD counter process time:

collected from observation and interviews over a period of four weeks. It is shown in table 4.

iii)

### Finance counter Process time:

collected from observation and interviews over a period of four weeks. It is shown in table 4.

iv)

### Doctors' process time:

collected from observation and interviews over a period of four weeks. The process time started as soon as the patient was called for treatment and ended as soon as the patient left the doctor's room.

### Maximum

Arrival rate / hour

33

35

44

Service rate for information counter / hour

60

30

20

Service rate for OPD counter / hour

6

8

10

Service rate of Finance / cashier counter / hour

120

60

40

Service rate of Clinic / hour per clinic

0.667

1.6

15

### Table - 6

We will take average values for our data analysis.

### Doctors'schedulingtimes:

Data was obtained from the Assistant manager (operations). There are ten clinics for different kind of diseases. In the morning session (9 am to 1 pm), each doctor has four hours of duty in which they have schedule of appointments. Each clinic has a standard number of patients but usually they are overbooked and people have to wait for their turn. Evening shift starts from 2pm to 5 pm.

### RESULTS:

The detailed results are as discussed below:

### Patients inter arrival data:

Using Arena input analyzer, we got inter arrival distribution.

### Data Summary

Distribution: Beta

Number of Data Points = 21

Square Error: 0.036109

Max Data Value = 30

Expression: 0.5 + 30 * BETA(0.554, 1.14)

Min Data Value = 1

Sample Mean = 10.3

### Histogram Summary

Sample Std Dev = 8.56

Histogram Range = 0.5 to 30.5

Number of Intervals = 30

### Patients'Waiting Time

Patient's waiting time is calculated which has maximum of 90 mins, while min is 4 minutes and average waiting time is 37 mins. While standard waiting as purposed in literature is 30 minutes.

### Doctors' Process Time:

Using the data with the help of Arena input Analyzer, following are some statistics;

### Data Summary

Expression: 3.5 + 87 * BETA(0.675, 0.909)

Number of Data Points = 18

Distribution: Beta

Max Data Value = 90

Square Error: 0.054106

Sample Std Dev = 26

Min Data Value = 4

Sample Mean = 37.1

### CONCLUSION

In studying the operations at SKMHT & RC, we find that OPD takes the longest time while the finance counter takes the least time. There are two queues (information queue and OPD Queue) that are unstable as (λ > µ), which are causing low utilization of Cashier counter and clinic counter which has 13% and 50% respectively.

In spite of having unstable queues, the patients have to wait for 37 minutes; If 30 minutes or below is set as the standard patients' waiting time, as suggested by Valdivia and Crowe (1997), then the patients' waiting time is unsatisfactory.

The utilization rate of information counter is 117 %. While server utilization of OPD counters is 375 %. It means both counter are unable to meet the patient demand, causing long queues, and delay in their treatment. The patients come on the day couldn't be able to get examined by the doctor. Patients have lot of time wasted in queues. Patients are given long dates for consultation. Since patients waiting time is unsatisfactory and doctors are under-utilized, several measures have to be taken by the hospital management.

### RECOMMENDATIONS

1. For any queue it is necessary it should be in steady state i.e. Service rate must be greater than the arrival rate (µ > λ). System should have capacity to handle the arrival of patients.

2. In case of OPD at SKMCH & RCH, there is a problem of capacity. As shown in data above, it's clear that information counter, OPD counters doesn't have capacity to cope with the load of patient. Hospital management is over utilizing these service counters. This is causing long waits for patients.

3. To handle these two issues, over utilization of service counters and long waiting time, there is need to increase the servers. So that they can not only save their service stations from over utilization but also reduce the patient wait time. Also they can be able to have steady state queues in that case.

4. Also some service stations are utilized by both process causing congestion and creates long queue. It is also suggested to separate these two phases, the registration phase and the appointment phase. In this way, we can reduce both waiting time for patients and server utilization.

5. Another suggestion to avoid the space occupation at OPD is to use telephone technology for patient registration and give them appointments on Phone.

6. Another way to minimize the waiting time at OPD clinic, Finance department can be pooled to OPD, which can perform both functions i.e. its own as well as of OPD.

7. The study was based on the data collected over a period of four weeks only. Therefore, the numbers might not be accurate. For better accuracy, the data should be collected for a period longer than four weeks.

8. As we have suggested about the increase in number of servers, it might cause an increase in the cost of new setup and it also shift more patients to clinics for examination and might cause over booking at doctors end. There is need to find the cost / benefit analysis for increasing the number of servers.

### Limitations:

Firstly the time allocated for this academic exercise does not permit data collection over a longer period of time and thus, may affect the representativeness of SKMTH patients. Moreover, the instructions laid down by the management of SKMTH eliminated the opportunity to conduct a survey on the flow of patients with assistance from the various service providers which would have provided a richer source of primary data for analysis.

Secondly, several assumptions are made in order to execute the analysis. This again affects the accuracy of the key operating statistics computed namely, the mean server utilization and expected waiting time.

### Future Research:

This academic research has been primarily concerned with the measures to improve the patient service level without incorporating the cost implication of the recommendations made, such as the addition of Servers. Hence a separate study to consider the cost aspects may be carried out to supplement the findings of this academic exercise.

If richer source of data can be obtained, other operations research techniques may be applied at various service stations.