Summary M&S2 Fundamentals of Modelling and Simulation

M&S 2
Statistics Workshops

M&S 2 – Statistics

Week Subjects
Week 1 – o Simulation.
Simulation by hand. o Components of a system.
o Queueing KPIs.
o Types of simulation.
o Steps in a simulation.
o Simulation by hand (Excel)
Week 2 – Queueing - Queueing Theory
models and Little’s - Kendall’s Notation on queues.
Law - Little’s Law (Excel)
- Queuing Formulas (Excel)
Week 3 – o Sample size versus Population
Hypothesis Testing, o Confidence interval mean/proportion (Excel)
Sample Size o Null and alternative hypothesis
o Errors in hypothesis testing
Week 4 – Goodness - Chi-square (X2) goodness of fit test
of fit - Assumptions
- 6 Steps of Hypothesis Testing
Week 5 – o T-test versus Z-test
Hypothesis testing, o T-test (student’s t-distribution)
one-sample T- and o Z-test
Z-test
Week 6 – Two - Dependent versus Independent
sample T-test - Two sample T-test for Independent samples
- Two sample T-test for dependent samples
Week 7 – Levenee’s o One-way ANOVA
and ANOVA test o Levene’s test
Week 8 – Terms
with definition

, M&S 2
Statistics Workshops


Week 1 – Simulation by hand
A simulation is the imitation of the operation of a real-world process
or system over time.
A system is defined as ‘a set of interrelated components working
together toward some common objective’.
A model is a simplified representation of the system of which we are
interested in study, modify it, predict its behavior or performance.

Components of a system:
 Entity: Object of interest in the system (customers, passengers, bags etc.).
 Attribute: Property of an entity (Schengen/Non-Schengen).
 Activity: Time period of specified length.
 State: Collection of variables to describe system (average waiting time, number of
desks in service, average queue length)
 Event: Instantaneous occurrence that may change state of system.

For example, components of the Check-in process at an airport:
Entity – Passengers
Attribute – # of bags
Activity – Checking In
State – Average waiting time
Event – Arrival at terminal

Queueing KPIs
For queueing problems, we need to determine four fundamental Key Performance Indicators
(KPIs).
W Average system time
WQ Average waiting time in queue
L Average number in system
LQ Average number in queue
ρ (rho) Average server utilization



P
P = Number of passengers served.
∑ System timeof passenger i served N = Number of passengers in the system.
i=1
w= T = Final clock time.
P
N

∑ Waiting time ∈queue Different Parameters
w Q= i=1 L LQ is the average arrival rate
N λ= = (expected number of arrivals per
W WQ
Areaunder L ( t ) graph unit time).
L=
T L is the average service rate of the
λ( ) server (expected number of
W
Areaunder LQ ( t ) graph μ= passengers completing service
L Q= ρ per time unit).
T
1 is the average service time (in
μ minutes).

, M&S 2
Statistics Workshops

Area under ρ ( t ) graph
ρ=
T



Output Performance Measures
 Average number of pax in queue at time t.
 Maximum number of pax in queue.




  Average and maximum total time
in system of pax (cycle time).
 Average waiting time of pax in queue.
 Maximum waiting time of pax in queue.
 Average utilization of the server (proportion of time busy).

Stop of Simulation
With stopping time Without stopping time
- Simulation ends at a predefined - Simulation ends when last passenger
stopping time. has been served.
- When P ≤ N - When P = N

Types of simulations – Discrete Event System (DES)
- Deterministic or Stochastic
o Some state variables are random.
- Static or Dynamic
o Time evolution is important.
- Continuous or Discrete
o System state changes when an event occurs.

Steps in a Simulation
Phase 1. – Initial phase
1. Problem formulation.
2. Setting objectives and overall project plan.
Phase 2. – Model building and data collection.
3. Model conceptualization.
4. Data collection.
5. Model translation.
6. Verification.
7. Validation.
Phase 3. – Run the model.
8. Experimental design.