Simulation and Modelling - Unit Wise Questions

Model

• Model is a computerized program that defines the mechanics of the considered system.
• During system modeling, the focus is on the system components, system structure, relationships among the components of the system and the behavior of the modeled system.
• Model must have state which may change on each time step.
• Model represents the system for the purpose of studying the system.

Types of Model

1. Physical Model

    Physical model is the smaller or larger physical copy of an object being modeled. The physical model helps in visualization of the object taken into consideration in an effective way. It is also used to solve equations with the particular boundary conditions.

Two types of physical model:

Simulation is the imitation of the operation of a real-world process or system over time. Simulation involves the generation of an artificial history of the system, and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system that is represented.

Steps in simulation study

1.Problem formulation

     Clearly state the problem.

2. Setting of objectives and overall project plan

     How we should approach the problem.

3. Model conceptualization

     Establish a reasonable model.

4. Data collection

    Collect the data necessary to run the simulation (such as arrival rate, arrival process, service discipline, service rate etc.).

5. Model translation

     Convert the model into a programming language.

6. Verification

     Verify the model by checking if the program works properly. Use common sense.

7. Validation

     Check if the system accurately represent the real system.

8. Experimental design

     How many runs? For how long? What kind of input variations?

9. Production runs and analysis

     Actual running the simulation, collect and analize the output.

10. Repeatition

     Repeat the experiments if necessary.

11. Document and report

     Document and report the results.

12. Implementation: Implement the simulated system in the real world if the simulation results show that it is advantageous to implement the new system or policy.

Simulation is the imitation of the operation of a real-world process or system over time. Simulation involves the generation of an artificial history of the system, and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system that is represented.

Steps in simulation study

1.Problem formulation

     Clearly state the problem.

2. Setting of objectives and overall project plan

     How we should approach the problem.

3. Model conceptualization

     Establish a reasonable model.

4. Data collection

    Collect the data necessary to run the simulation (such as arrival rate, arrival process, service discipline, service rate etc.).

5. Model translation

     Convert the model into a programming language.

6. Verification

     Verify the model by checking if the program works properly. Use common sense.

7. Validation

     Check if the system accurately represent the real system.

8. Experimental design

     How many runs? For how long? What kind of input variations?

9. Production runs and analysis

     Actual running the simulation, collect and analyze the output.

10. Repetition

     Repeat the experiments if necessary.

11. Document and report

     Document and report the results.

12. Implementation: Implement the simulated system in the real world if the simulation results show that it is advantageous to implement the new system or policy.

Limitations of simulation 

Analogy between Mechanical system and electrical system 

Consider the two systems shown in following figures i.e. Figure 1 and Figure 2.

Fig1: Mechanical System

Fig2: Electrical system

The Figure 1. represents a mass that is subject to an applied force F(t) varying with time, a spring whose force is proportional to its extension or contraction, and a shock absorber that exerts a damping force proportional to the velocity of the mass.It can be shown that the motion of the system is described by the following differential equation.

Where,

x is the distance moved, M is the mass, K is the stiffness of the spring & D is the damping factor of the shock absorber.

Figure 2. represents an electrical circuit with an inductance L, a resistance R, and a capacitance C, connected in series with a voltage source that varies in time according to the function E(t). If q is the charge on the capacitance, it can be shown that the behavior of the circuit is governed by the following differential equation:

Inspection of these two equations shows that they have exactly the same form and that the following equivalences occur between the quantities in the two systems:

a) Displacement x = Charge q
b) Velocity x’ = Current I, q’
c) Force F = Voltage E
d) Mass M = Inductance L
e) Damping Factor D = Resistance R
f) Spring stiffness K = Inverse of Capacitance 1/C
g) Acceleration x’’ = Rate of change of current q’’

The mechanical system and the electrical system are analogs of each other, and the performance of either can be studied with the other.

Static mathematical model is the mathematical model that represents the logical view of the system in equilibrium state. Such models are time-invariant. It is generally represented by the basic algebraic equations.

For example:  Supply and demand relationship model of a market 

Consider a market model assuming the linear behavior of supply and demand with price.

• Demand for a product will be low when the price is high and will be high when the price is low.
• Supply for a product will be high when the price is high and will be low when the price is low.
• If conditions remain stable, the price will settle to the point at which demand equals supply.

The relations can be stated mathematically as:    

    Q = a – bP
    S = c + dP
    S = Q

where, S = supply, Q = demand and P = price

The linear market model is shown below:

Difference Between Stochastic and Deterministic Activities

• Stochastic model / activity has one or more random variable as inputs. Random inputs leads to random outputs. E.g. Simulation of a bank involves random inter-arrival and service times.
• Deterministic model / activity contains no random variables. They have a known set of inputs which will result in a unique set of outputs. E.g. Arrival of patients to the Dentist at the scheduled appointment time.

A group of components which are interconnected and interact to fulfill certain objective is called a system. A system mainly consists of 4 elements: input, process, output and feedback.

        Fig: Block diagram of system

Input: It involves capturing and assembling elements that enter the system to be processed. Inputs to the system are anything to be captured by the system from its environment. For example, raw materials.

Processing: It involves transformation processes that convert input to output. For example, a manufacturing process.

Output: It involves transferring elements that have been produced by a transformation process to their ultimate destinations. Outputs are the things produced by the system and sent into its environment. For example, finished products.

Feedback:  It is the idea of monitoring the current system output and comparing it to the system goal. Any variation from the goal are then fed back in to the system and used to adjust it to ensure that it meets its goal. For example, data about sales performance is feedback to a sales manager.

A system boundary is a line that separates the system from the system environment. A change occuring outside the system may affect the performance of the system. Such changes are said to occur in the system environment. Therefore the system environment means anything outside the system boundary that may affect the system.

Components of the system

1. Entity: An entity is an object of interest for the purpose of the system study.

2. Attribute: The property or the characteristics of an entity is called attribute.

3. Activity: The time period of specified length is called activity. The activity changes the state of the system.

4. State: The state of a system is defined as the collection of variables necessary to describe the system at any time with respect to objectives of the study.

5. Event: The instantaneous occurance that may change the state of a system is called event. There are 2 types of event:

    a) Endogenous Event: The event that occurs within the system and change the state of the system is called endogenous event.

    b) Exogenous Event: The event that occurs outside the system but affects the state of the system is called exogenous event.

Simulation is the imitation of the operation of a real-world process or system over time. Simulation involves the generation of an artificial history of the system, and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system that is represented.

Advantages of Simulation

- New procedures or rules can be explored without hampering the ongoing real time system operations.
- New designs can be tested without acquiring resources.
- The hypotheses about certain phenomena can be tested for feasibility.
- Insights can be obtained about the variable interaction and their credit for system performance.
- It provides the better understanding of the operation of the system in easier way.

- A simulation study can help in understanding how the system operates rather than how individuals think the system operates.

- “what-if” questions can be answered. So it is useful in the design of new systems.

Disadvantages of Simulation

- Building model requires specialization which is the art learned through experience.
- In some cases, the results of the simulation may be so complex to interpret.
- It may be time consuming and expensive.

Simulation is the imitation of the operation of a real-world process or system over time. Simulation involves the generation of an artificial history of the system, and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system that is represented.

The following lists when simulation is useful:

• Simulation enables the study of, and experimentation with, the internal interactions of a complex system, or a subsystem within a complex system.
• Informational, organizational, and environmental changes can be simulated, and the effects of these alterations on the model’s behavior can be observed.
• Simulation can be used as a pedagogical device to reinforce analytic solution methodologies.
  
• Simulation can be used to experiment with new designs or policies prior to implementation
• Simulation can be used to verify analytic solutions.
• By simulating different capabilities for a machine, requirements can be determined.
• Animation shows a system in simulated operations so that the plan can be visualized.
• The modern system is so complex that the interactions can be treated only through simulation.

The following lists when simulation is not useful:

• Simulation is not appropriate when the problem can be solved by common sense.
• Simulation should not be used if the problem can be solved analytically.
• Simulation should not be used if it is easier to perform direct implementations.
• Simulation should not be used if the cost exceeds the savings.
• Simulation should not be done if the resources and time are not available.
• Simulation can not be done if no data is available.
• Simulation is not preferred if the system behavior is too complex to be defined.

Model is a computerized program that defines the mechanics of the considered system. It must have state which may change on each time step. Model represents the system for the purpose of studying the system.

A simulation model can be classified as being static or dynamic, deterministic or stochastic, and discrete or continuous.

• A static simulation model, sometimes called a Monte Carlo simulation, represent a system at a particular point in time. For example, the profit of a company, which is affected by 3 variables drawn from different statistics distributions.
• A dynamic simulation model represents a system as it changes over time. For example, the simulation of a bank from 9 am to 4 pm.
• Deterministic model contains no random variables. They have a known set of inputs which will result in a unique set of outputs. E.g. Arrival of patients to the Dentist at the scheduled appointment time.
• Stochastic model has one or more random variable as inputs. Random inputs leads to random outputs. E.g. Simulation of a bank involves random inter-arrival and service times.
• Discrete model is one in which the state variables changes only at a discrete set of time. For example: banking system in which no of customers (state variable) changes only when a customer arrives or service provided to customer i.e. customer depart form system.
• Continuous model is one in which the state variables change continuously over time. For example, during winter seasons level of which water decreases gradually and during rainy season level of water increase gradually. The change in water level is continuous.

A group of components which are interconnected and interact to fulfill certain objective is called a system. A system mainly consists of input, process, output and feedback.

        Fig: Block diagram of system

A system boundary is a line that separates the system from the system environment. A change occuring outside the system may affect the performance of the system. Such changes are said to occur in the system environment. Therefore the system environment means anything outside the system boundary that may affect the system.

Components of the system

1. Entity: An entity is an object of interest for the purpose of the system study.

2. Attribute: The property or the characteristics of an entity is called attribute.

3. Activity: The time period of specified length is called activity. The activity changes the state of the system.

4. State: The state of a system is defined as the collection of variables necessary to describe the system at any time with respect to objectives of the study.

5. Event: The instantaneous occurance that may change the state of a system is called event. There are 2 types of event:

    a) Endogenous Event: The event that occurs within the system and change the state of the system is called endogenous event.

    b) Exogenous Event: The event that occurs outside the system but affects the state of the system is called exogenous event.

Phases of Simulation Study

I Phase:

• Consists of steps 1 and 2
• It is period of discovery/orientation
• The analyst may have to restart the process if it is not fine-tuned
• Recalibrations and clarifications may occur in this phase or another phase.

II Phase:

• Consists of steps 3,4,5,6 and 7
• A continuing interplay is required among the steps
• Exclusion of model user results in implications during implementation

III Phase:

• Consists of steps 8,9 and 10
• Conceives a thorough plan for experimenting
• Discrete-event stochastic is a statistical experiment
• The output variables are estimates that contain random error and therefore proper statistical analysis is required.

IV Phase:

• Consists of steps 11 and 12
• Successful implementation depends on the involvement of user and every steps successful completion.

Phases of Simulation Study

I Phase:

• Consists of steps 1 and 2
• It is period of discovery/orientation
• The analyst may have to restart the process if it is not fine-tuned
• Recalibrations and clarifications may occur in this phase or another phase.

II Phase:

• Consists of steps 3,4,5,6 and 7
• A continuing interplay is required among the steps
• Exclusion of model user results in implications during implementation

III Phase:

• Consists of steps 8,9 and 10
• Conceives a thorough plan for experimenting
• Discrete-event stochastic is a statistical experiment
• The output variables are estimates that contain random error and therefore proper statistical analysis is required.

IV Phase:

• Consists of steps 11 and 12
• Successful implementation depends on the involvement of user and every steps successful completion.

Simulation is the imitation of the operation of a real-world process or system over time. Simulation involves the generation of an artificial history of the system, and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system that is represented.

Merits of Simulation

• New procedures or rules can be explored without hampering the ongoing real time system operations.
• New designs can be tested without acquiring resources.
• The hypotheses about certain phenomena can be tested for feasibility.
• Insights can be obtained about the variable interaction and their credit for system performance.
• It provides the better understanding of the operation of the system in easier way.
• A simulation study can help in understanding how the system operates rather than how individuals think the system operates.
• “what-if” questions can be answered. So it is useful in the design of new systems.

Demerits of Simulation

• Building model requires specialization which is the art learned through experience.
• In some cases, the results of the simulation may be so complex to interpret.
• It may be time consuming and expensive.

Interactive systems are computer systems characterized by significant amounts of interaction between humans and the computer. Editors, CAD-CAM (Computer Aided Design-Computer Aided Manufacture) systems, and data entry systems are all computer systems involving a high degree of human-computer interaction. Games and simulations are interactive systems. Web browsers and Integrated Development Environments (IDEs) are also examples of very complex interactive systems.

The earliest interactive systems were command line systems, which tightly controlled the interaction between the human and the computer. The user was required to know the commands that might be issued and how the arguments were to be ordered. Both the UNIX operating system and DOS (Disk Operating System) are classic examples. Users were required to enter data in a particular sequence. The options for the output of data were also tightly controlled, and generally limited. Such systems generally put a high demand on the user to remember commands and the syntax for issuing these commands.

An interactive system has the potential for variation and unpredictability in its response, and depending on the context may well be considered more in terms of a composition or structured improvisation rather than an instrument. 

Activity: The time period of specified length is called activity.

Event: An instantaneous occurrence that might change the state of the system is called event.

State variables: A collection of variables necessary to describe a system at any time is called state variables.

Now,

a. Supermarkets

    Activities:  sell product, fill displays, organize the supermarket carts

    Events: Arrival and departure of customers, reception of products, lack of products

b. Inventory control

   Activities:  Withdrawing

    Events: Demand

c. Hospital

    Activities:  vaccinate, feed the patient, administering medicines, heal the wounds

    Events: lack of medicines, lack of staff, lack of beds, arrival and departure of patients.

Models that have the property of changing only at fixed interval of time is called distributed lag model. It is used to predict current values of a dependent variable based on both the current values of an explanatory variable (independent variable) and the lagged (past period) values of this explanatory variable.

This model consists of linear algebraic equations that represent continuous system but data are available at fixed points in time.

For example: Mathematical model of national economy

Let

C = consumption

I = investment

T = Taxes

G = government expenditures

Y = national income

Then

C=20+0.7(Y-T)

I=2+0.1Y

T=0.2Y

Y=C+I+G

All the equation are expressed in billions of rupees. This is static model and can be made dynamic by lagging all the variables as follows

C=20+0.7(Y_-1-T_-1)

I=2+0.1Y_-1

T=0.2Y_-1

Y=C_-1+I_-1+G_-1

Any variable that can be expressed in the form of its current value and one or more previous value is called lagging variable. And hence this model is given the name distributed lag model. The variable in a previous interval is denoted by attaching –n suffix to the variable. Where –n indicate the nth interval.

Phases of Simulation Study

I Phase:

• Consists of steps 1 and 2
• It is period of discovery/orientation
• The analyst may have to restart the process if it is not fine-tuned
• Recalibrations and clarifications may occur in this phase or another phase.

II Phase:

• Consists of steps 3,4,5,6 and 7
• A continuing interplay is required among the steps
• Exclusion of model user results in implications during implementation

III Phase:

• Consists of steps 8,9 and 10
• Conceives a thorough plan for experimenting
• Discrete-event stochastic is a statistical experiment
• The output variables are estimates that contain random error and therefore proper statistical analysis is required.

IV Phase:

• Consists of steps 11 and 12
• Successful implementation depends on the involvement of user and every steps successful completion.

Explain...

Models that have the property of changing only at fixed interval of time is called distributed lag model. It is used to predict current values of a dependent variable based on both the current values of an explanatory variable (independent variable) and the lagged (past period) values of this explanatory variable.

This model consists of linear algebraic equations that represent continuous system but data are available at fixed points in time.

For example: Mathematical model of national economy

Let

C = consumption

I = investment

T = Taxes

G = government expenditures

Y = national income

Then

C=20+0.7(Y-T)

I=2+0.1Y

T=0.2Y

Y=C+I+G

All the equation are expressed in billions of rupees. This is static model and can be made dynamic by lagging all the variables as follows

C=20+0.7(Y_-1-T_-1)

I=2+0.1Y_-1

T=0.2Y_-1

Y=C_-1+I_-1+G_-1

Any variable that can be expressed in the form of its current value and one or more previous value is called lagging variable. And hence this model is given the name distributed lag model. The variable in a previous interval is denoted by attaching –n suffix to the variable. Where –n indicate the nth interval.

Explain....

a. Cafeteria

    Entities:  customer, cashier, waiter

    Attributes: hungry, hurried

     Activities: take orders, collect payments, dispatch orders

    Events: arrival and departure of customer, lack of raw material, faults in the equipment

    State variables: number of occupied tables, number of orders per table, number of customers waiting for attention

b. Inventory

    Entities:  Warehouse

    Attributes:Capacity

   Activities:  Withdrawing

    Events: Demand

    State variables: Levels of inventory, background

c. Banking

    Entities: Customers

    Attributes: Checking-account balance

    Activities:  Making deposits

    Events: Arrival, departure

    State variables: Number of busy tellers, number of customer waiting

d. A hospital emergency room

     Entities:  doctor, nurse, patient, equipment

    Attributes:safe, expert in his/her science, intelligent, responsible

    Activities:  vaccinate, feed the patient, administering medicines, heal the wounds

    Events: lack of medicines, lack of staff, lack of beds, arrival and departure of patients.

    State variables: number of patients, amount of administrated medicines, number of empty beds, number of registerd cases.

e. Communication

    Entities: Messages

    Attributes: Length, destination

    Activities: Transmitting

    Events: Arrival at destination

    State variables: Number waiting to be transmitted

Models that have the property of changing only at fixed interval of time is called distributed lag model. It is used to predict current values of a dependent variable based on both the current values of an explanatory variable (independent variable) and the lagged (past period) values of this explanatory variable.

This model consists of linear algebraic equations that represent continuous system but data are available at fixed points in time.

For example: Mathematical model of national economy

Let

C = consumption

I = investment

T = Taxes

G = government expenditures

Y = national income

Then

C=20+0.7(Y-T)

I=2+0.1Y

T=0.2Y

Y=C+I+G

All the equation are expressed in billions of rupees. This is static model and can be made dynamic by lagging all the variables as follows

C=20+0.7(Y_-1-T_-1)

I=2+0.1Y_-1

T=0.2Y_-1

Y=C_-1+I_-1+G_-1

Any variable that can be expressed in the form of its current value and one or more previous value is called lagging variable. And hence this model is given the name distributed lag model. The variable in a previous interval is denoted by attaching –n suffix to the variable. Where –n indicate the nth interval.

a. System, boundary and system environment

• A system is defined as a collection of object join in some regular interaction or interdependency for achievement of a common goal. We can also define a system as organized set of inter-related idea or principles. All system have input, output and feedback and maintain a basic level of equilibrium.
• A system boundary is the line that separate the system and its environment.
• The external components which interact with the system and produce necessary changes are said to constitute the system environment. In modeling systems, it is necessary to decide on the boundary between the system and its environment. This
  
  

b. Real time simulation

Real-time simulation refers to a computer model of a physical system that can execute at the same rate as actual "wall clock" time. In other words, the computer model runs at the same rate as the actual physical system.

For example, if a tank takes 10 minutes to fill in the real-world, the simulation would take 10 minutes as well. Real-time simulation occurs commonly in computer gaming, but also is important in the industrial market for operator training and off-line controller tuning.

a. Banking

    Entities: Customers

    Attributes: Checking-account balance

    Activities:  Making deposits

    Events: Arrival, departure

b. Communication

    Entities: Messages

    Attributes: Length, destination

    Activities: Transmitting

    Events: Arrival at destination

c. Production

    Entities: Machines

    Attributes: Speed, capacity, breakdown rate

    Activities: welding, stamping

    Events: Breakdown

d. Inventory

    Entities:  Warehouse

    Attributes: Capacity

   Activities:  Withdrawing

    Events: Demand

    State variables: Levels of inventory, background

e. Supermarkets

    Entities: Customers, cashier, salesperson

    Attributes: Faster, reliable, responsible, punctual, skilled in sales

    Activities:  sell product, fill displays, organize the supermarket carts

    Events: Arrival and departure of customers, reception of products, lack of products

a) System and its environment.

• A system is defined as a collection of object join in some regular interaction or interdependency for achievement of a common goal. We can also define a system as organized set of inter-related idea or principles. All system have input, output and feedback and maintain a basic level of equilibrium.
• A system boundary is the line that separate the system and its environment.
• The external components which interact with the system and produce necessary changes are said to constitute the system environment. In modeling systems, it is necessary to decide on the boundary between the system and its environment. This

b) Simulation run statistics

Analog computers are those computers that are unified with devices like adder and integral so as to simulate the continuous mathematical model of the system, which generates continuous outputs.

Analog method of system simulation is for use of analog computer and other analog devices in the simulation of continuous system. The analog computation is sometimes called differential analyser. Electronics analog computers for simulation are based on the use of high gain dc amplifiers called operational amplifier (op amps). In such analog computer, voltages are equated to mathematical variables and the op amps can add and integrate the voltages. The proper configurations can handle addition of several input voltages each representing the input variables. The analog computer provides limited accuracy because op amps have many assumptions which can never be true in reality.

The general method to apply analog computers for the simulation of continuous system models involves following components:

Example: Automobile Suspension Problem

The general method by which analog computers are applied can be demonstrated using second order differential equation.

    M x" + D x' + K x = K F(t)

Solving the equation for the highest order derivate gives,

    M x" = K F(t) – D x' - K x

        Fig: Automobile suspension problem

Suppose a variable representing the input F(t) is supplied, assume there exist variables representing -x and -x'. These three variables can be scaled and added to produce Mx". Integrating it with a scale factor 1/M produces X'. Changing sign produces -x', further integrating produces -x, a further sign inverter is included to produce +x as output.

Analog simulation and Digital simulation differs in the following ways:

• The goal of digital simulation is much simpler to achieve than analog simulation because the circuit is discrete in nature and has no circuit constraints to meet, such as Kirchhoff's Current Law (KCL).
• Digital simulation runs orders of magnitude faster than analog simulation because digital simulation deals with high-level behavior only, while in analog simulation the same elements have analog implementations.
• Digital simulation abstracts away important electrical characteristics that might be revealed by the analog simulation.

Analog Method of simulation

Analog computers are those computers that are unified with devices like adder and integral so as to simulate the continuous mathematical model of the system, which generates continuous outputs.

Analog method of system simulation is for use of analog computer and other analog devices in the simulation of continuous system. The analog computation is sometimes called differential analyser. Electronics analog computers for simulation are based on the use of high gain dc amplifiers called operational amplifier (op amps). In such analog computer, voltages are equated to mathematical variables and the op amps can add and integrate the voltages. The proper configurations can handle addition of several input voltages each representing the input variables. The analog computer provides limited accuracy because op amps have many assumptions which can never be true in reality.

The general method to apply analog computers for the simulation of continuous system models involves following components:

Example: Automobile Suspension Problem

The general method by which analog computers are applied can be demonstrated using second order differential equation.

    M x" + D x' + K x = K F(t)

Solving the equation for the highest order derivate gives,

    M x" = K F(t) – D x' - K x

        Fig: Automobile suspension problem

Suppose a variable representing the input F(t) is supplied, assume there exist variables representing -x and -x'. These three variables can be scaled and added to produce Mx". Integrating it with a scale factor 1/M produces X'. Changing sign produces -x', further integrating produces -x, a further sign inverter is included to produce +x as output.

Discrete system is one in which the state variables changes only at a discrete set of time. For example: banking system in which no of customers (state variable) changes only when a customer arrives or service provided to customer i.e. customer depart form system.

Continuous system is one in which the state variables change continuously over time. For example, the head of the water behind a dam. During and for some time after a rainstorm, water flows into the lake behind the dam. Water is drawn from the dam for flood control and to make electricity. Evaporation also decreases the water level.

--> Discrete model is appropriate for system having discrete state and changes at particular time point and remains in that state for some time where as Continuous model is a appropriate for system with a continuous state that changes continuously over time.

--> Most discrete time system represents how discrete signals are transformed via difference equations where as Most continuous time system represent how continuous signals are transferred via differential equations.

--> A discrete simulation model is not always used to model a discrete system, nor is a continuous model always used to model a continuous system.

Analog computers are those computers that are unified with devices like adder and integral so as to simulate the continuous mathematical model of the system, which generates continuous outputs.

Analog method of system simulation is for use of analog computer and other analog devices in the simulation of continuous system. The analog computation is sometimes called differential analyser. Electronics analog computers for simulation are based on the use of high gain dc amplifiers called operational amplifier (op amps). In such analog computer, voltages are equated to mathematical variables and the op amps can add and integrate the voltages. The proper configurations can handle addition of several input voltages each representing the input variables. The analog computer provides limited accuracy because op amps have many assumptions which can never be true in reality.

Now, 

Designing analog computer for Automobile Suspension system:

The general method by which analog computers are applied can be demonstrated using second order differential equation.

    M x" + D x' + K x = K F(t)

Solving the equation for the highest order derivate gives,

    M x" = K F(t) – D x' - K x

        Fig: Automobile suspension problem

Suppose a variable representing the input F(t) is supplied, assume there exist variables representing -x and -x'. These three variables can be scaled and added to produce Mx". Integrating it with a scale factor 1/M produces X'. Changing sign produces -x', further integrating produces -x, a further sign inverter is included to produce +x as output.

Interactive System

Interactive systems are computer systems characterized by significant amounts of interaction between humans and the computer. Editors, CAD-CAM (Computer Aided Design-Computer Aided Manufacture) systems, and data entry systems are all computer systems involving a high degree of human-computer interaction. Games and simulations are interactive systems. Web browsers and Integrated Development Environments (IDEs) are also examples of very complex interactive systems.

Feedback System

The system takes feedback from the output i.e. input is coupled with output. A significant factor in the performance of many systems is that coupling occurs between the input and output of the system. The term feedback is used to describe the phenomenon.

One example of feedback system in which there is continuous control is the aircraft system. Here the input is a desired aircraft heading and the output is the actual heading. The gyroscope of the autopilot is able to detect the difference between the two headings. A feedback is established by using the difference to operate the control surface. Since change of heading will then affect the signal being used to control the heading.

The difference between the desired signal θ_t and actual heading θ₀ is called the error signal, since it is a measure of the extent to which the system from the desired condition. It is denoted by є.

We also know that, in terms of angular acceleration

From equation (1), (2) & (3)

Dividing both sides by I, and making the following substitutions in equation (4)

 (where  is damping factor)

    This is a second order differential equation.

Clock time is updated based on the following two models:

1. Fixed time-step model

In this the timer simulated by the computer is updated at a fixed time interval. The system is checked to see if any event has taken place during that interval. All the events which take place during the time interval are considered to have occurred simultaneously at the end of the interval.

2. Event-to-event model

It is also known as the next-event model. In this the computer advances the time to the occurrence of the next event. So it shifts from one event to the next event and the system state does not change in between. A track of the current time is kept when something interesting happens to the system.

The flowcharts for both models is shown below.

The non-stationary Poisson process is a Poisson process for which the arrival rate varies with time. More specifically, it can be defined as follows:

The counting process N(t) is a non-stationary Poisson process if:

a) The process has independent increments.

b) 

    Where, λ(t) = the arrival rate at time t

                dt = differential sized interval

The definition is identical to the stationary Poisson process, with the exception that the arrival rate, λ(t), is now a function of time.

A counting process N(t) is a stationary Poisson process with rate  λ if

a) The process has independent increments.

b) The process has stationary increments. and

c) 

A non-stationary Poisson process can be transformed into a stationary Poisson process with arrival rate 1.

Arrival defines the way customers enter the system. Mostly the arrivals are random with random intervals between two adjacent arrivals. Typically the arrival is described by a random distribution of intervals also called Arrival Pattern. Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution and most important model for random arrival is the poisson process. In schedule arrival interarrival time of customers are constant.

The non-stationary Poisson process is a Poisson process for which the arrival rate varies with time. More specifically, it can be defined as follows:

The counting process N(t) is a non-stationary Poisson process if:

a) The process has independent increments.

b) 

    Where, λ(t) = the arrival rate at time t

                dt = differential sized interval

Fixed time-step model

In this the timer simulated by the computer is updated at a fixed time interval. The system is checked to see if any event has taken place during that interval. All the events which take place during the time interval are considered to have occurred simultaneously at the end of the interval.

Event-to-event model

It is also known as the next-event model. In this the computer advances the time to the occurrence of the next event. So it shifts from one event to the next event and the system state does not change in between. A track of the current time is kept when something interesting happens to the system.

The flowcharts for both models is shown below.

In reality, the system is of neither a pure continuous nor a pure discrete nature. For simulating such system, the combination of analog and digital computers are used. Such setup is known as hybrid computers. The simulation provided by the hybrid computers is known as hybrid simulation.

The form taken by hybrid simulation depends upon the application. One computer may be simulating the system being studied while other is providing a simulation of the environment in to which the system is to operate. It is also possible that the system being simulated is an interconnection of continuous and discrete subsystem, which can best be modeled by an analog and digital computer being linked together.. 

The major difficulty in use of hybrid simulation is that it requires high speed converters to transform signals from analog to digital form and vice versa.

The system takes feedback from the output i.e. input is coupled with output. A significant factor in the performance of many systems is that coupling occurs between the input and output of the system. The term feedback is used to describe the phenomenon.

One example of feedback system in which there is continuous control is the aircraft system. Here the input is a desired aircraft heading and the output is the actual heading. The gyroscope of the autopilot is able to detect the difference between the two headings. A feedback is established by using the difference to operate the control surface. Since change of heading will then affect the signal being used to control the heading.

The difference between the desired signal θ_t and actual heading θ₀ is called the error signal, since it is a measure of the extent to which the system from the desired condition. It is denoted by є.

We also know that, in terms of angular acceleration

From equation (1), (2) & (3)

Dividing both sides by I, and making the following substitutions in equation (4)

 (where  is damping factor)

    This is a second order differential equation.

Fixed time-step model

In this the timer simulated by the computer is updated at a fixed time interval. The system is checked to see if any event has taken place during that interval. All the events which take place during the time interval are considered to have occurred simultaneously at the end of the interval.

Event-to-event model

It is also known as the next-event model. In this the computer advances the time to the occurrence of the next event. So it shifts from one event to the next event and the system state does not change in between. A track of the current time is kept when something interesting happens to the system.

The flowcharts for both models is shown below.

Let us take an example, where CPU is running for 6 seconds and now it performs a calculation for 0.01 second and stops the calculation, again the CPU runs for 5 more seconds.

So, Clock time= total time CPU is active =6+5 secs = 11 seconds

Simulation time = Time taken by CPU to perform calculation = 0.01 seconds

Clock time is updated based on the following two models:

1. Fixed time-step model

In this the timer simulated by the computer is updated at a fixed time interval. The system is checked to see if any event has taken place during that interval. All the events which take place during the time interval are considered to have occurred simultaneously at the end of the interval.

2. Event-to-event model

It is also known as the next-event model. In this the computer advances the time to the occurrence of the next event. So it shifts from one event to the next event and the system state does not change in between. A track of the current time is kept when something interesting happens to the system.

The flowcharts for both models is shown below.

a. Discrete system modeling

In discrete systems, the changes in the system state are discontinuous and each change in the state of the system is called an event. The model used in a discrete system simulation has a set of numbers to represent the state of the system, called as a state descriptor. 

b. Feedback system

The system takes feedback from the output i.e. input is coupled with output. A significant factor in the performance of many systems is that coupling occurs between the input and output of the system. The term feedback is used to describe the phenomenon.

One example of feedback system in which there is continuous control is the aircraft system. Here the input is a desired aircraft heading and the output is the actual heading. The gyroscope of the autopilot is able to detect the difference between the two headings. A feedback is established by using the difference to operate the control surface. Since change of heading will then affect the signal being used to control the heading.

The difference between the desired signal θ_t and actual heading θ₀ is called the error signal, since it is a measure of the extent to which the system from the desired condition. It is denoted by є.

We also know that, in terms of angular acceleration

From equation (1), (2) & (3)

Dividing both sides by I, and making the following substitutions in equation (4)

 (where  is damping factor)

    This is a second order differential equation.

Characteristics of Queueing System

1. Calling Population

The population of potential customers those require service from system is called calling population. It may be finite or infinite. System having large calling population is usually considered as infinite. For e.g. customers at banks, restaurant. And System having less and countable population is usually considered as finite. For e.g. a certain number of machines to be repaired by a service man.

    In finite population model, arrival rate depends on the number of customers being served and waiting. But in infinite population model, arrival rate is not affected by the number of customer being served and waiting.

2. Arrival Process

The arrival process for infinite-population models is usually characterized in terms of interarrival times of successive customers. Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution and most important model for random arrival is the poisson process. In schedule arrival interarrival time of customers are constant.

3. Service Process

Service process can be measured by the number of customers served per some unit of time or the time taken to complete the service. Once entities have entered to the system they must be served. The service can be provided in single or batch. if it is batch, as in the case of arrival the batch size can be fixed or random. Service time may be of constant duration or of random duration.

Markov Service process: A Markov service process is a special service process in which entities are processed one at a time in FCFS order and service times are independent and exponential. Aswith the case of Markov arrivals, a Markov service process is memoryless, which means that the expected time until an entity is finished remains constant regardless of how long it has been in service.

4. Queueing Discipline and Queueing Behaviour

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first.

Queue behavior refers to the actions of customers while in a queue waiting for service to begin. Different queue behaviours are:

• Balk/Balking: It means leaving the queue when the customer see the line is too long.
• Renege/Reneging: Leave after being in the line when they see that the line is moving too slowly.
• Jockey/Jockeying: Move from one to a shortest line. 

5. Number of Servers: 

Servers represent the entity that provides service to the customer. A system may consist of single server or multiple servers.
- A system with multiple servers is able to provide parallel services to the customers.

The various performance measures in single server (M/M/1) queuing system simulation is discussed below:

1. If  Ta and Ts be the inter arrival time and the mean service time then

• Arrival rate λ=1/Ta 
• Service rate μ=1/Ts

2. The ratio of the mean service time to the mean inter arrival time is called traffic intensity. i.e.  u=Ts/Ta

3. Server utilization: It consists of only the arrival that gets served. It is denoted by and defined as

        ρ= λTs= λ/ μ (server utilization for single server).

4. Probability of finding service counter free is (1 – ρ) i.e there are zero customers in the service facility.

The line where the entities or customer wait is generally known as queue. The combination of all entities in system being served and being waiting for service will be called as queueing system.

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first. 

The performance of a queuing system can be evaluated in terms of a number of response parameters, however the following four are generally employed.

• Average number of customer in the queue or in the system
• Average waiting time of the customer in the queue or in the system
• System utilization (Server utilization)
• The cost of waiting time and idle time

1. If  Ta and Ts be the inter arrival time and the mean service time then

• Arrival rate λ=1/Ta 
• Service rate μ=1/Ts
• Average number of customer in the system = λ/(μ - λ)
• Average number of customer in the queue = λ2/μ(μ - λ)
• Average waiting time in the system = 1/(μ - λ)
• Average waiting time in the queue = λ/μ(μ - λ)

2. The ratio of the mean service time to the mean inter arrival time is called traffic intensity. i.e.  u=Ts/Ta

3. Server utilization: It consists of only the arrival that gets served. It is denoted by and defined as

        ρ= λTs= λ/ μ (server utilization for single server).

        ρ= λTs= λ/ nμ (server utilization for n server). 

4. Probability of finding service counter free is (1 – ρ) i.e there are zero customers in the service facility.

The line where the entities or customer wait is generally known as queue. The combination of all entities in system being served and being waiting for service will be called as queueing system.

Characteristics/Elements of Queueing System

The key elements of queuing systems are customer and server. Customer refers to anything that arrives at a facility and requires service. E.g. people, machines, trucks, emails. Servers refers to any resource that provides the requested service. E.g. receptionist, tellor, CPU, washing machine etc.

1. Calling Population

The population of potential customers those require service from system is called calling population. It may be finite or infinite. System having large calling population is usually considered as infinite. For e.g. customers at banks, restaurant. And System having less and countable population is usually considered as finite. For e.g. a certain number of machines to be repaired by a service man.

    In finite population model, arrival rate depends on the number of customers being served and waiting. But in infinite population model, arrival rate is not affected by the number of customer being served and waiting.

2. Arrival Process

The arrival process for infinite-population models is usually characterized in terms of interarrival times of successive customers. Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution and most important model for random arrival is the poisson process. In schedule arrival interarrival time of customers are constant.

3. Service Process

Service process can be measured by the number of customers served per some unit of time or the time taken to complete the service. Once entities have entered to the system they must be served. The service can be provided in single or batch. if it is batch, as in the case of arrival the batch size can be fixed or random. Service time may be of constant duration or of random duration.

Markov Service process: A Markov service process is a special service process in which entities are processed one at a time in FCFS order and service times are independent and exponential. Aswith the case of Markov arrivals, a Markov service process is memoryless, which means that the expected time until an entity is finished remains constant regardless of how long it has been in service.

4. Queueing Discipline and Queueing Behaviour

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first. 

Queue behavior refers to the actions of customers while in a queue waiting for service to begin. Different queue behaviours are:

• Balk/Balking: It means leaving the queue when the customer see the line is too long.
• Renege/Reneging: Leave after being in the line when they see that the line is moving too slowly.
• Jockey/Jockeying: Move from one to a shortest line. 

5. Number of Servers: 

Servers represent the entity that provides service to the customer. A system may consist of single server or multiple servers.
- A system with multiple servers is able to provide parallel services to the customers.

The line where the entities or customer wait is generally known as queue. The combination of all entities in system being served and being waiting for service will be called as queueing system.

Characteristics of Queueing System

The key elements of queuing systems are customer and server. Customer refers to anything that arrives at a facility and requires service. E.g. people, machines, trucks, emails. Servers refers to any resource that provides the requested service. E.g. receptionist, tellor, CPU, washing machine etc.

1. Calling Population

The population of potential customers those require service from system is called calling population. It may be finite or infinite. System having large calling population is usually considered as infinite. For e.g. customers at banks, restaurant. And System having less and countable population is usually considered as finite. For e.g. a certain number of machines to be repaired by a service man.

    In finite population model, arrival rate depends on the number of customers being served and waiting. But in infinite population model, arrival rate is not affected by the number of customer being served and waiting.

2. Arrival Process

The arrival process for infinite-population models is usually characterized in terms of interarrival times of successive customers. Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution and most important model for random arrival is the poisson process. In schedule arrival interarrival time of customers are constant.

3. Service Process

Service process can be measured by the number of customers served per some unit of time or the time taken to complete the service. Once entities have entered to the system they must be served. The service can be provided in single or batch. if it is batch, as in the case of arrival the batch size can be fixed or random. Service time may be of constant duration or of random duration.

Markov Service process: A Markov service process is a special service process in which entities are processed one at a time in FCFS order and service times are independent and exponential. Aswith the case of Markov arrivals, a Markov service process is memoryless, which means that the expected time until an entity is finished remains constant regardless of how long it has been in service.

4. Queueing Discipline and Queueing Behaviour

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first.

Queue behavior refers to the actions of customers while in a queue waiting for service to begin. Different queue behaviours are:

• Balk/Balking: It means leaving the queue when the customer see the line is too long.
• Renege/Reneging: Leave after being in the line when they see that the line is moving too slowly.
• Jockey/Jockeying: Move from one to a shortest line. 

5. Number of Servers: 

Servers represent the entity that provides service to the customer. A system may consist of single server or multiple servers.
- A system with multiple servers is able to provide parallel services to the customers.

A congestion system is system in which there is a demand for resources for a system, and when the resources become unavailable, those requesting the resources wait for them to become available. The level of congestion in such systems is usually measured by the waiting line, or queue, of resource requests (waiting line or queuing models).

Characteristics/Elements of Queueing System

The key elements of queuing systems are customer and server. Customer refers to anything that arrives at a facility and requires service. E.g. people, machines, trucks, emails. Servers refers to any resource that provides the requested service. E.g. receptionist, tellor, CPU, washing machine etc.

1. Calling Population

The population of potential customers those require service from system is called calling population. It may be finite or infinite. System having large calling population is usually considered as infinite. For e.g. customers at banks, restaurant. And System having less and countable population is usually considered as finite. For e.g. a certain number of machines to be repaired by a service man.

    In finite population model, arrival rate depends on the number of customers being served and waiting. But in infinite population model, arrival rate is not affected by the number of customer being served and waiting.

2. Arrival Process

The arrival process for infinite-population models is usually characterized in terms of interarrival times of successive customers. Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution and most important model for random arrival is the poisson process. In schedule arrival interarrival time of customers are constant.

3. Service Process

Service process can be measured by the number of customers served per some unit of time or the time taken to complete the service. Once entities have entered to the system they must be served. The service can be provided in single or batch. if it is batch, as in the case of arrival the batch size can be fixed or random. Service time may be of constant duration or of random duration.

Markov Service process: A Markov service process is a special service process in which entities are processed one at a time in FCFS order and service times are independent and exponential. Aswith the case of Markov arrivals, a Markov service process is memoryless, which means that the expected time until an entity is finished remains constant regardless of how long it has been in service.

4. Queueing Discipline and Queueing Behaviour

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first.

Queue behavior refers to the actions of customers while in a queue waiting for service to begin. Different queue behaviours are:

• Balk/Balking: It means leaving the queue when the customer see the line is too long.
• Renege/Reneging: Leave after being in the line when they see that the line is moving too slowly.
• Jockey/Jockeying: Move from one to a shortest line. 

5. Number of Servers: 

Servers represent the entity that provides service to the customer. A system may consist of single server or multiple servers.
- A system with multiple servers is able to provide parallel services to the customers.

A congestion system is system in which there is a demand for resources for a system, and when the resources become unavailable, those requesting the resources wait for them to become available. The level of congestion in such systems is usually measured by the waiting line, or queue, of resource requests (waiting line or queuing models).

Characteristics/Elements of Queueing System

The key elements of queuing systems are customer and server. Customer refers to anything that arrives at a facility and requires service. E.g. people, machines, trucks, emails. Servers refers to any resource that provides the requested service. E.g. receptionist, tellor, CPU, washing machine etc.

1. Calling Population

The population of potential customers those require service from system is called calling population. It may be finite or infinite. System having large calling population is usually considered as infinite. For e.g. customers at banks, restaurant. And System having less and countable population is usually considered as finite. For e.g. a certain number of machines to be repaired by a service man.

    In finite population model, arrival rate depends on the number of customers being served and waiting. But in infinite population model, arrival rate is not affected by the number of customer being served and waiting.

2. Arrival Process

The arrival process for infinite-population models is usually characterized in terms of interarrival times of successive customers. Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution and most important model for random arrival is the poisson process. In schedule arrival interarrival time of customers are constant.

3. Service Process

Service process can be measured by the number of customers served per some unit of time or the time taken to complete the service. Once entities have entered to the system they must be served. The service can be provided in single or batch. if it is batch, as in the case of arrival the batch size can be fixed or random. Service time may be of constant duration or of random duration.

Markov Service process: A Markov service process is a special service process in which entities are processed one at a time in FCFS order and service times are independent and exponential. Aswith the case of Markov arrivals, a Markov service process is memoryless, which means that the expected time until an entity is finished remains constant regardless of how long it has been in service.

4. Queueing Discipline and Queueing Behaviour

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first. 

Queue behavior refers to the actions of customers while in a queue waiting for service to begin. Different queue behaviours are:

• Balk/Balking: It means leaving the queue when the customer see the line is too long.
• Renege/Reneging: Leave after being in the line when they see that the line is moving too slowly.
• Jockey/Jockeying: Move from one to a shortest line. 

5. Number of Servers: 

Servers represent the entity that provides service to the customer. A system may consist of single server or multiple servers.
- A system with multiple servers is able to provide parallel services to the customers.

A congestion system is system in which there is a demand for resources for a system, and when the resources become unavailable, those requesting the resources wait for them to become available. The level of congestion in such systems is usually measured by the waiting line, or queue, of resource requests (waiting line or queuing models).

Characteristics/Elements of Queueing System

The key elements of queuing systems are customer and server. Customer refers to anything that arrives at a facility and requires service. E.g. people, machines, trucks, emails. Servers refers to any resource that provides the requested service. E.g. receptionist, tellor, CPU, washing machine etc.

1. Calling Population

The population of potential customers those require service from system is called calling population. It may be finite or infinite. System having large calling population is usually considered as infinite. For e.g. customers at banks, restaurant. And System having less and countable population is usually considered as finite. For e.g. a certain number of machines to be repaired by a service man.

    In finite population model, arrival rate depends on the number of customers being served and waiting. But in infinite population model, arrival rate is not affected by the number of customer being served and waiting.

2. Arrival Process

The arrival process for infinite-population models is usually characterized in terms of interarrival times of successive customers. Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution and most important model for random arrival is the poisson process. In schedule arrival interarrival time of customers are constant.

3. Service Process

Service process can be measured by the number of customers served per some unit of time or the time taken to complete the service. Once entities have entered to the system they must be served. The service can be provided in single or batch. if it is batch, as in the case of arrival the batch size can be fixed or random. Service time may be of constant duration or of random duration.

4. Queueing Discipline and Queueing Behaviour

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

Queue behavior refers to the actions of customers while in a queue waiting for service to begin. 

5. Number of Servers: 

Servers represent the entity that provides service to the customer. A system may consist of single server or multiple servers.
- A system with multiple servers is able to provide parallel services to the customers.

The key elements of queuing systems are customer and server. Customer refers to anything that arrives at a facility and requires service. E.g. people, machines, trucks, emails. Servers refers to any resource that provides the requested service. E.g. receptionist, tellor, CPU, washing machine etc.

1. Calling Population

The population of potential customers those require service from system is called calling population. It may be finite or infinite. System having large calling population is usually considered as infinite. For e.g. customers at banks, restaurant. And System having less and countable population is usually considered as finite. For e.g. a certain number of machines to be repaired by a service man.

    In finite population model, arrival rate depends on the number of customers being served and waiting. But in infinite population model, arrival rate is not affected by the number of customer being served and waiting.

2. Arrival Process

The arrival process for infinite-population models is usually characterized in terms of interarrival times of successive customers. Arrivals may occur at scheduled times or at random times. When at random times, the inter arrival times are usually characterized by a probability distribution and most important model for random arrival is the poisson process. In schedule arrival interarrival time of customers are constant.

3. Service Process

Service process can be measured by the number of customers served per some unit of time or the time taken to complete the service. Once entities have entered to the system they must be served. The service can be provided in single or batch. if it is batch, as in the case of arrival the batch size can be fixed or random. Service time may be of constant duration or of random duration.

4. Queueing Discipline and Queueing Behaviour

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

Queue behavior refers to the actions of customers while in a queue waiting for service to begin.

5. Number of Servers: 

Servers represent the entity that provides service to the customer. A system may consist of single server or multiple servers.
- A system with multiple servers is able to provide parallel services to the customers.

The multi server queue consists of multiple servers and a common queue for all items. When any item requests for the server, it is allocated if at-least one server is available. Else the queue begins to start until the server is free. In this system, we assume that all servers are identical, i.e. there is no difference which server is chosen for which item.

The total server utilization for N server system is 

ρ = λ/μN

    where, μ = service rate and  λ = arrival rate

                        Fig: Multi server queue

Server/System utilization is the percentage of the time that all servers are busy. System utilization factor (ρ ) is the ratio of average arrival rate (λ) to the average service rate (μ).

ρ = λ/μ in the case of a single server model

ρ = λ/μn in the case of a “n” server model

The system utilization can be increased by increasing the arrival rate which amounts to increasing the average queue length as well as the average waiting time, as shown in fig below. Under the normal circumstances 100% system utilization is not a realistic goal.

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first. 

Traffic Intensity

The ratio of the mean service time to the mean inter arrival time is called traffic intensity. I.e. u= λ"T_s or u=T_s/T_a 

Server Utilization

Server utilization is the percentage of the time that all servers are busy. Server utilization factor (ρ ) is the ratio of average arrival rate (λ) to the average service rate (μ).

ρ = λ/μ in the case of a single server model

ρ = λ/μn in the case of a “n” server model

Kendall's notation

Kendall’s notation is used for parallel server systems. The basic format of this notation is of form: A / B / c / N / K, where, A, B, c, N, K respectively indicate arrival pattern, service pattern, number of servers, system capacity, and Calling population.

The symbols used for the probability distribution for inter arrival time, and service time are, D for deterministic, M for exponential (or Markov) and Ek for Erlang.

E.g. 

a) M/D/2/5/∞ stands for a queuing system having exponential arrival times, deterministic service time, 2 servers, capacity of 5 customers, and infinite population.

b) M/D/2 means exponential arrival time, deterministic service time, 2 servers, infinite service capacity, and infinite population.

Kendall’s notation is used for parallel server systems. The basic format of this notation is of form: A / B / c / N / K, where, A, B, c, N, K respectively indicate arrival pattern, service pattern, number of servers, system capacity, and Calling population.

The symbols used for the probability distribution for inter arrival time, and service time are, D for deterministic, M for exponential (or Markov) and Ek for Erlang.

E.g. 

a) M/D/2/5/∞ stands for a queuing system having exponential arrival times, deterministic service time, 2 servers, capacity of 5 customers, and infinite population.

b) M/D/2 means exponential arrival time, deterministic service time, 2 servers, infinite service capacity, and infinite population.

Traffic Intensity

The ratio of the mean service time to the mean inter arrival time is called traffic intensity. I.e. u= λ"T_s or u=T_s/T_a 

Server Utilization

Server utilization is the percentage of the time that all servers are busy. Server utilization factor (ρ ) is the ratio of average arrival rate (λ) to the average service rate (μ).

ρ = λ/μ in the case of a single server model

ρ = λ/μn in the case of a “n” server model

Kendall's notation

Kendall’s notation is used for parallel server systems. The basic format of this notation is of form: A / B / c / N / K, where, A, B, c, N, K respectively indicate arrival pattern, service pattern, number of servers, system capacity, and Calling population.

The symbols used for the probability distribution for inter arrival time, and service time are, D for deterministic, M for exponential (or Markov) and Ek for Erlang.

E.g. 

a) M/D/2/5/∞ stands for a queuing system having exponential arrival times, deterministic service time, 2 servers, capacity of 5 customers, and infinite population.

b) M/D/2 means exponential arrival time, deterministic service time, 2 servers, infinite service capacity, and infinite population.

The line where the entities or customer wait is generally known as queue. The combination of all entities in system being served and being waiting for service will be called as queueing system.

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first.

Now,

Mean arrival rate = λ

Mean inter-arrival time = T_a

Total working hours = 5 * 8 = 40 hrs per week

Total calls in the week = 1200

T_a  =  (40*60*60 sec)/1200 = 144,000 sec/1200 = 120 sec = 2 min

λ = 1/T_a = 1/2 = 0.5 call per min

a. Queueing Discipline

Queue discipline refers to the rule that a server uses to choose the next customer from the queue when the server completes the service of the current customer. Common queue disciplines include first-in-first-out (FIFO); last-in-first-out (LIFO); service in random order (SIRO); shortest processing time first (SPT); and service according to priority (PR).

• First in first out :This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
• Last in first out : This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. 
• Service in random order: A customer is picked up randomly from the waiting queue for service.
• Shortest job first: The next job to be served is the one with the smallest size (shortest service time).
• Priority: Customers with high priority are served first.

b. CSMP

CSMP or Continuous System Modelling Program is an early scientific computer software designed for modelling and solving differential equations numerically. This enables real-world systems to be simulated and tested with a computer.

Types of statements in CSMP:

1. Structural statements which defines the model. They consist of FORTARN like statement and functional block designed for operations that frequently occur in a model definition. Structural statement can make use of the operation of addition, subtraction, multiplication, division and exponentiation, using the same notation and rule as are used in FORTRAN. If the model include the equation .Then the following statement would be used

         X=6.0*Y/W+(Z-2)**2.0

2. Data statements which assign numerical value to parameters, constant and initial conditions. For example one data statement called INCON can be used to set the initial value of integration function block.

3. Control statements which specify options in the assembly and execution of program and choice of inputs. For e.g. If printed output is required, control statements with PRINT and PRDEL are used followed by the names of variables to form the outputs.

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc.

Formally,

A Markov chain is a sequence of random variables X₁, X₂, X₃, ... with the Markov property, namely that, given the present state, the future and past states are independent.

The conditional probability above gives us the probability that a process in state i_n at time n moves to i_n+1 at time n + 1. We call this the transition probability for the Markov chain. If the transition probability does not depend on the time n, we have a stationary Markov chain, with transition probabilities

Now we can write down the whole Markov chain as a matrix P:

Example:

Weather Problem:

• raining today  ⇒ 40 % rain tomorrow

                                ⇒ 60 % no rain tomorrow

• not raining today  ⇒ 20 % rain tomorrow

                                      ⇒ 80 % no rain tomorrow

What will be probability if todays is not raining then not rain the day after tomorrow?

Markov chain diagram:

Transition matrix:

Thus, the probability of not rainy the day after tomorrow is 0.76.

Applications of Markov Chain

1. Physics: Markovian systems appear extensively in thermodynamics and statistical mechanics, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description.

2. Internet applications: The Page Rank of a webpage as used by Google is defined by a Markov chain. It is the probability to be at page i in the stationary distribution on the following Markov chain on all (known) web pages

3. Statistics: Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called Markov chain Monte Carlo (MCMC) And many more.

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc.

Formally,

A Markov chain is a sequence of random variables X₁, X₂, X₃, ... with the Markov property, namely that, given the present state, the future and past states are independent.

The conditional probability above gives us the probability that a process in state i_n at time n moves to i_n+1 at time n + 1. We call this the transition probability for the Markov chain. If the transition probability does not depend on the time n, we have a stationary Markov chain, with transition probabilities

Now we can write down the whole Markov chain as a matrix P:

Example:

Weather Problem:

• raining today  ⇒ 40 % rain tomorrow

                                ⇒ 60 % no rain tomorrow

• not raining today  ⇒ 20 % rain tomorrow

                                      ⇒ 80 % no rain tomorrow

What will be probability if todays is not raining then not rain the day after tomorrow?

Markov chain diagram:

Transition matrix:

Thus, the probability of not rainy the day after tomorrow is 0.76.

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc.

Formally,

A Markov chain is a sequence of random variables X₁, X₂, X₃, ... with the Markov property, namely that, given the present state, the future and past states are independent.

The conditional probability above gives us the probability that a process in state i_n at time n moves to i_n+1 at time n + 1. We call this the transition probability for the Markov chain. If the transition probability does not depend on the time n, we have a stationary Markov chain, with transition probabilities

Now we can write down the whole Markov chain as a matrix P:

Example:

Weather Problem:

• raining today  ⇒ 40 % rain tomorrow

                                ⇒ 60 % no rain tomorrow

• not raining today  ⇒ 20 % rain tomorrow

                                      ⇒ 80 % no rain tomorrow

What will be probability if todays is not raining then not rain the day after tomorrow?

Markov chain diagram:

Transition matrix:

Thus, the probability of not rainy the day after tomorrow is 0.76.

Applications of Markov Chain

1. Physics: Markovian systems appear extensively in thermodynamics and statistical mechanics, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description.

2. Internet applications: The Page Rank of a webpage as used by Google is defined by a Markov chain. It is the probability to be at page i in the stationary distribution on the following Markov chain on all (known) web pages

3. Statistics: Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called Markov chain Monte Carlo (MCMC) And many more.

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc.

Key features of Markov chains

1. The outcome of each experiment is one of a set of discrete state.

2. The outcome of the experiment depends only on the present state and not on the past state.

3. The transition probability remains constant from one to the next.

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc.

Applications of Markov Chain

1. Physics: Markovian systems appear extensively in thermodynamics and statistical mechanics, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description.

2. Internet applications: The Page Rank of a webpage as used by Google is defined by a Markov chain. It is the probability to be at page i in the stationary distribution on the following Markov chain on all (known) web pages

3. Statistics: Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called Markov chain Monte Carlo (MCMC) And many more.

4. Queuing theory: Markov chains are the basis for the analytical treatment of queues (queuing theory). Agner Krarup Erlang initiated the subject in 1917. This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources (such as bandwidth).

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc.

Formally,

A Markov chain is a sequence of random variables X₁, X₂, X₃, ... with the Markov property, namely that, given the present state, the future and past states are independent.

The conditional probability above gives us the probability that a process in state i_n at time n moves to i_n+1 at time n + 1. We call this the transition probability for the Markov chain. If the transition probability does not depend on the time n, we have a stationary Markov chain, with transition probabilities

Now we can write down the whole Markov chain as a matrix P:

Example:

Weather Problem:

• raining today  ⇒ 40 % rain tomorrow

                                ⇒ 60 % no rain tomorrow

• not raining today  ⇒ 20 % rain tomorrow

                                      ⇒ 80 % no rain tomorrow

What will be probability if todays is not raining then not rain the day after tomorrow?

Markov chain diagram:

Transition matrix:

Thus, the probability of not rainy the day after tomorrow is 0.76.

a. Markov Chain

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc.

b. Feedback system

The system takes feedback from the output i.e. input is coupled with output. A significant factor in the performance of many systems is that coupling occurs between the input and output of the system. The term feedback is used to describe the phenomenon.

One example of feedback system in which there is continuous control is the aircraft system. Here the input is a desired aircraft heading and the output is the actual heading. The gyroscope of the autopilot is able to detect the difference between the two headings. A feedback is established by using the difference to operate the control surface. Since change of heading will then affect the signal being used to control the heading.

The difference between the desired signal θ_t and actual heading θ₀ is called the error signal, since it is a measure of the extent to which the system from the desired condition. It is denoted by є.

We also know that, in terms of angular acceleration

From equation (1), (2) & (3)

Dividing both sides by I, and making the following substitutions in equation (4)

 (where  is damping factor)

    This is a second order differential equation.

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc.

Formally,

A Markov chain is a sequence of random variables X₁, X₂, X₃, ... with the Markov property, namely that, given the present state, the future and past states are independent.

The conditional probability above gives us the probability that a process in state i_n at time n moves to i_n+1 at time n + 1. We call this the transition probability for the Markov chain. If the transition probability does not depend on the time n, we have a stationary Markov chain, with transition probabilities

Now we can write down the whole Markov chain as a matrix P:

Example:

Weather Problem:

• raining today  ⇒ 40 % rain tomorrow

                                ⇒ 60 % no rain tomorrow

• not raining today  ⇒ 20 % rain tomorrow

                                      ⇒ 80 % no rain tomorrow

What will be probability if todays is not raining then not rain the day after tomorrow?

Markov chain diagram:

Transition matrix:

Thus, the probability of not rainy the day after tomorrow is 0.76.

a. Differential equation

The equation that consists of the higher order derivatives of the dependent variable is known as differential equations.
- The differential equation is said to be linear if any of the dependent variables and its derivatives have power of one and are multiplied by the constant.
E.g. M x’’ + D x’ + K x = K F(t)
where, M, D and K are constants; F(t) is the input to the system depending upon the independent variable t; x’’ and x’ are second and first order derivatives of dependent variable x.
- The differential equation is said to be non-linear if the dependent variable or any of its derivatives are raised to a power or are combined in other way like multiplication.

The differential equation is said to be partial if more than one independent variables occur in a differential equation.
-E.g. Equation of flow of heat in three dimensional body. It consists of four independent variables ( three dimensions and time ) and one dependent variable ( temperature ).

b. Markov Chain

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc).

Difference between true and pseudo random numbers

• Pseudo random numbers are the random numbers that are generated by using some known methods (algorithms) so as to produce a sequence of numbers in [0,1] that can simulates the ideal properties of random numbers. They are not completely random as the set of random numbers can be replicated because of use of some known method.
• True random numbers are gained from physical processes like radioactive decay or also rolling a dice and introduce it into a computer.
• Pseudo random numbers have fast response  in generating numbers while true random have slow response.
• In pseudo random number, sequence of numbers can be reproduced where as In true random number, sequence of numbers can't be reproduced.
• In pseudo random number, sequence of number is repeated where as In true random number, sequence of numbers will or will not be repeated.
  

Properties of Random Numbers

1. Uniformity:

• The random numbers generated should be uniform. That means a sequence of random numbers should be equally probable every where.
• If we divide all the set of random numbers into several numbers of class interval then number of samples in each class should be same.
• If ‘N’ number of random numbers are divided into ‘K’ class interval, then expected number of samples in each class should be equal to e_i = N / K.

2. Independent:

• Each random number should be independent samples drawn from a continuous uniform distribution between 0 and 1.
• The probability density function is given by:
  f(x) = 1, 0 <= x <= 1
        = 0, otherwise

3. Maximum Density:

• The large samples of random number should be generated in a given range.

4. Maximum Cycle:

•  It states that the repetition of numbers should be allowed only after a large interval of time.

Now,

Given sequence of number,

    0.37, 0.29, 0.19, 0.88 0.44, 0.63, 0.77, 0.70 0.21, and 0.58 

Arranging the given number in ascending order:

    0.19, 0.21, 0.29, 0.37, 0.44, 0.58, 0.63, 0.7, 0.77, 0.88

Here, N = 10

Calculation table for Kolmogorov-Smirnov test :

i
1	0.19	0.1	-	0.19
2	0.21	0.2	-	0.11
3	0.29	0.3	0.01	0.09
4	0.37	0.4	0.03	0.07
5	0.44	0.5	0.06	0.04
6	0.58	0.6	0.02	0.08
7	0.63	0.7	0.07	0.03
8	0.7	0.8	0.1	-
9	0.77	0.9	0.13	-
10	0.88	1.0	0.12	-