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.
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.
