THEORY EXAMINATION (SEM–VI) 2016-17 MODELING AND SIMULATION

MODELING AND SIMULATION (NIT062)

SECTION – A

(Attempt All | 10 × 2 = 20 Marks)

(a) Actual system vs Simulation

An actual system is a real-world system that exists physically, whereas simulation is an abstract or mathematical representation of that system used to study its behavior without disturbing the real system.

(b) Purpose of Dynamic Modeling

Dynamic modeling is used to represent time-dependent behavior of systems where system variables change with time, such as population growth or inventory systems.

(c) Simulating gender of children using random numbers

Assume:

Random number 0–49 → Male

Random number 50–99 → Female

By generating random numbers, the gender of children can be simulated assuming equal probability.

(d) Why simulation is required?

Simulation is required when:                         Real system experimentation is costly or risky

Analytical solutions are not possible             System is complex and stochastic

Time compression or expansion is needed

(e) Outcomes of Cobweb Model

Cobweb model outcomes include:               Convergent oscillations

Divergent oscillations                                   Continuous oscillations

Stable equilibrium

(f) Flow chart for next-event simulation

Main steps:                                                   Initialize system state

Advance simulation clock to next event      Execute event

Update system state                                    Schedule future events

Check termination condition                       (Drawn neatly in exam.)

(g) Stochastic simulation

Stochastic simulation involves random variables and probabilistic behavior.
Example: Queuing system with random arrivals and service times.

(h) Builder selling houses problem

As unsold houses decrease, selling rate decreases → negative feedback system.
This represents a dynamic simulation problem solved using differential equations or system dynamics.

(i) Advantages & disadvantages of simulation languages

Advantages:                                              Faster model development

Built-in random generators                        Easy statistical analysis

Disadvantages:                                         Limited flexibility

Requires learning specific syntax               Higher execution time

(j) Important feature in project planning & control

Critical Path is the most important feature.
It is resolved using CPM (Critical Path Method) to identify minimum project duration.

SECTION – B

(Attempt Any Five | 5 × 10 = 50 Marks)

(a) Components of Discrete Event System Model      Main components:

Entities: Customers, jobs                                           Attributes: Properties of entities

Activities: Time-consuming operations                    Events: Instantaneous occurrences

State variables: Describe system state                     Queues: Waiting lines

Example: Bank queuing system.

(b) Classification of Models & Limitations

Classification:                                                          Physical vs Mathematical

Static vs Dynamic                                                      Deterministic vs Stochastic

Continuous vs Discrete

Limitations:                                                             Approximation of reality

Requires assumptions                                               May ignore human behavior

(c) Fixed time-step vs Next-event model

(d) Monte Carlo Simulation & Area under sine curve

Steps:                                                                      Define problem

Identify probability distribution                               Generate random numbers

Perform experiment                                                 Analyze results

Area under sin(x) from 0 to π:                             Exact value = 2
Monte Carlo estimates area by random sampling.

(e) Exponential growth & decay models

Growth:

dXdt=kX⇒X=X0ekt\frac{dX}{dt} = kX \Rightarrow X = X_0 e^{kt}dtdX​=kX⇒X=X0​ekt

Decay:

dXdt=−kX⇒X=X0e−kt\frac{dX}{dt} = -kX \Rightarrow X = X_0 e^{-kt}dtdX​=−kX⇒X=X0​e−kt

Examples: Population growth, radioactive decay.

(f) ATM machine simulation (Single-server queue)   Arrivals → Poisson distribution

Service → Exponential distribution                           One server (ATM)

Performance measures:                                             Average waiting time

Queue length                                                            Server utilization

(g) CPM for given activities

Steps:                                                                        Draw network diagram

Forward pass → earliest times                                  Backward pass → latest times

Identify critical path (zero slack)                               Critical path determines minimum project duration.

(h) Simulation languages & selection                      Examples: GPSS, SIMSCRIPT, SIMULA

Selection depends on:                                           Type of system

Model complexity                                                     User expertise

Output & visualization needs

SECTION – C

(Attempt Any Two | 2 × 15 = 30 Marks)

 (a) Casino dice fairness test

Observed data for 100 trials:

Using Chi-square test, calculated χ² > critical value →
Conclusion: Dice are not fair.

(b) Pseudo random number generation

Methods:                                                       Linear Congruential Method

Xn+1=(aXn+c)mod  mX_{n+1} = (aX_n + c) \mod mXn+1​=(aXn​+c)modm

Middle square method                                 Properties:

Uniformity                                                    Independence

Long period

Simulation of Water Reservoir System

Steps:                                                           Identify inflow & outflow

Define storage capacity                               Use mass balance equation

Storaget+1=Storaget+Inflow−OutflowStorage_{t+1} = Storage_t + Inflow - OutflowStoraget+1​=Storaget​+Inflow−Outflow

Simulate over time                                       Analyze overflow and shortage

Applications: dams, irrigation planning.

 Bakery shop demand simulation

Demand distribution:

Random numbers:

21, 27, 47, 54, 60, 39, 43, 91, 25, 20

Simulated demand (10 days):

25, 25, 35, 35, 35, 35, 35, 40, 25, 25