THEORY EXAMINATION (SEM–IV) 2016-17 BASICS OF SYSTEM MODELING AND SIMULATION

✅ SECTION – A (Very Short Answers) 

a) Define modelling.
Modelling is creating a simplified representation of a real system for study and analysis.

b) What are discrete variables?
Variables that change only at specific points or take integer values.

c) What is a data model?
A structured representation of input data and its relationships used in a system or simulation.

d) What is distributed lag?
When the effect of an input variable is spread across several future time periods.

e) What is CDF?
CDF is the cumulative distribution function: F(x)=P(X≤x)F(x) = P(X \le x)F(x)=P(X≤x).

f) Define simulators.
Software or hardware that imitates the behaviour of a real system.

g) What is service delay?
The time taken to actually serve a customer (service time).

h) What is parameter data?
Fixed numerical values that define model behaviour (like λ, μ, capacity).

i) Define error.
Difference between actual value and estimated/model value.

j) Define quenching (queuing) system.
A system where customers arrive, wait in a queue, get service, and leave.

SECTION – B (Only short / relevant explanations)

(a) Properties of CDF (short)

Non-decreasing

F(−∞)=0,  F(+∞)=1F(-\infty)=0,\; F(+\infty)=1F(−∞)=0,F(+∞)=1

Right-continuous

Probability between points = F(b)−F(a)F(b)-F(a)F(b)−F(a)

(b) Steps for creating system modelling (short)

Problem definition             System boundary

Conceptual model             Data collection

Model building                  Verification & validation

Experimentation                 Documentation

(c) Simulation – needs / advantages / disadvantages (short)

Simulation: imitation of real system behaviour.
Needs: complex systems, costly real experiments.
Advantages: flexible, safe testing, handles randomness.
Disadvantages: time consuming, costly, no direct optimal solution.
Three simulators: HYSYS, ASPEN Plus, CHEMCAD.

(d) Components of a system (short)

Entities, attributes, resources, events, activities, state variables, inputs, outputs.

(e) Multiple server queuing simulation (short)

Schedule arrival & departure events

If server free → service starts

Else → join queue

On service completion → next from queue

Collect: waiting time, queue length, utilization, etc.

(f) Monte Carlo method (short)

Random sampling technique to estimate numerical results.
Example: estimating π using random points inside a circle.

(g) Criteria for selecting modelling technique (short)

System complexity             Type of data

Accuracy needed               Cost & time

Software availability           Ease of understanding

(h) Characteristics of queuing system (short)

Arrival pattern, service mechanism, queue discipline, capacity, population size, number of servers.

SECTION – C (Minimal but exam-ready)

Q3. Poisson & exponential distribution (short explanation + Poisson derivation)

Poisson distribution:

P(X=k)=e−λλkk!P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}P(X=k)=k!e−λλk​

Exponential distribution:

f(t)=λe−λt,t≥0f(t)=\lambda e^{-\lambda t},\quad t\ge0f(t)=λe−λt,t≥0

Derivation of Poisson from Binomial:
Start with

P(X=k)=(nk)pk(1−p)n−kP(X=k)=\binom{n}{k}p^k(1-p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k

Let np=λnp=\lambdanp=λ, n→∞n\to\inftyn→∞, p→0p\to0p→0.
Then

P(X=k)=λkk!e−λ.P(X=k)=\frac{\lambda^k}{k!}e^{-\lambda}.P(X=k)=k!λk​e−λ. 

Q4. Goodness-of-Fit tests (only names + short meaning)

Chi-Square Test: compares observed vs expected frequencies.

K-S Test: compares empirical CDF vs theoretical CDF.

Anderson–Darling: like K-S but gives more weight to tails.

Q5. Short notes (only key points)

(i) Computer network model:
Simulates packets, nodes, routing, delays, throughput.

(ii) Optimization:
Finding best solution within constraints (max/min objective).

(iii) Capital recovery model:
Annual value A=P⋅i(1+i)n(1+i)n−1A = P \cdot \frac{i(1+i)^n}{(1+i)^n-1}A=P⋅(1+i)n−1i(1+i)n​.

(iv) Job shop model:
Jobs follow different routes through machines; focus on scheduling.

(v) π value estimation:
Monte Carlo: π ≈ 4 × (points inside circle / total points).