Delay Analysis Workbook
Air Transportation Systems
Engineering
Delay Analysis Workbook
1
Copyright Lance Sherry 2010
Delay Analysis Workbook
Air Transportation Delay Analysis Workbook
Actions:
1.
Read Chapter 23 Flows and Queues at Airports
2.
Answer the following questions.
Introduction
Page 821
A generic queuing system consists of three components:
1.
_____________________________________________
2.
_____________________________________________
3.
_____________________________________________
Each user of the queuing system is generated by the ______________________, passes
through the _______________ where it remains for some time period (zero to t), and then
is processed by one of the parallel ____________.
A queuing network is a set of _________________________________________. In a
queuing network, the user sources for some of the queuing systems maybe
__________________________________________________.
User Generation Process
Page 822
Two properties of the User Generation Process:
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Delay Analysis Workbook
1.
The ____________________________ is the rate at which the users arrive
over time. The Greek letter __________ is used to denote this parameter.
Example demand rate: The rate at which flights arrive at La Guardia airport during the
peak period of operations is 8 flights per 15 minutes.
2.
The ___________________________ is the time interval between successive
demands. This time interval is referred to as the
____________________________ or shortened to ________________
Example probability distributions: (1) the flights arrive at the runway evenly spaced (i.e.
one flight every 90 seconds), (2) the flight arrive on average once every 90 seconds, but
distributed randomly (and independently) in time.
The probability distribution for the 1st example is known as ________________ (or
__________________) demand.
The probability distribution for the 2nd example is known as
___________________________________
The process of random, independent distribution is known as the
______________________ Process. This process exhibits a negative exponential
probability density function.
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Draw a negative exponential probability density function. The x-axis is Demand Rate
Inter-arrival Time, the y-axis is Probability Density
Short demand inter-arrival time occur with high/low probability. (Circle one).
Long-demand inter-arrival times occur with high/low probability (Circle one)
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The expected value (average) length of the demand inter-arrival is equal to 1/λ which is
equal to the _________________ demand rate.
Describe in qualitative terms what happens with Poisson probability distributions.
Which type of system is more likely to experience delays – arrival rate with deterministic
inter-arrival time distribution, or an arrival rate with Poisson inter-arrival time
distribution __________________________________________?
Explain why____________________________________________________________
________________________________________________________________________
________________________________________________________________________
Aside: Why do the Washington DC Metro stations have 2 escalators from platform-tostreet and only 1 escalator from street-to-platform? Explain.
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Delay Analysis Workbook
Service Processes
Page 823
The service rate is the ____________________________________ per unit time. This
parameter is represented by the Greek letter ________________.
Example. The service rate for a runway is 12 flights per 15 minutes in Visual
Meteorological Conditions (VMC) and 10 flights in Instrument Meteorological
Conditions.
The probability distribution that describes the ___duration___ of service times is
known as _____________________________
Example: The probability distribution for service times of a runway is known as Runway
Occupancy Time (ROT). ROT varies by aircraft type. A typical ROT has µ= 45 seconds,
and σ = 8 secs.
Queuing Process
Page 826
Most queues for aircraft at airports operate on a First-Come/First-Serve (FCFS) discipline.
Explain FCFS ____________________________________________________________
________________________________________________________________________
________________________________________________________________________
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Constrained Position Switch (CPS) is an alternate queuing discipline. Explain what
CPS is and how it works.
_____________________________________________________________________
A crucial parameter in describing and designing airport queuing systems is queue
capacity. This is the ____________________________________ ________________
Examples of queue capacity for flights:
1. Departure Runway queues are limited by ________________________________
_____________________________________________________________________
2. Arrival Runway queues are limited by __________________________________
____________________________________________________________________
Measures of Performance and Level of Service
Page 828+
Utilization Ratio, also known as _______________________ ratio.
Represented by Greek letter ______________________
Utilization Ratio is defined as ___________ / _____________
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When ρ > 1 system is considered to be ______________________________ and delays
are _______________________
When ρ = 1, delays are __________________________
When ρ < .78, delays are __________________________
Waiting Time, represented by ___________, is defined as ________________________
________________________________________________________________________
Number of Users in the queue, represented by ___________, is defined as ___________
________________________________________________________________________
Waiting Time and Number of Users in queue are ___________________ variables
because demand interval times and service times are _____________________________
As a consequence the Expected Values of these parameters are used.
E[Wq] is the ____________________________________________________________
E[Nq] is the _____________________________________________________________
The most common measure of variability in delay is the variance in Wq represented as
__________________________________ or ________________________________
A large variance or standard deviation indicates high/low (circle on) variability in delay.
To combat high variability flight schedules incorporate _slack time__ in their
schedules. Explain how slack time addresses the issue of variability _________________
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Delay Analysis Workbook
________________________________________________________________________
Reliability is defined as the probability ________________________________________
An example of a reliability metric is the percentage of flights that arrive with 15 minutes
of the scheduled arrival time.
Draw a log-normal distribution with a long right tail representing flight delays. Sketch
in the region of the distribution representing the “late” flights.
Explain where the rationale for the 15 minute threshold …
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
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Maximum Queue Length is used by designers to determine the amount of space (e.g.
taxiway lengths for departure queue) that should be provided to handle the maximum
queue. This parameter is measure of the risk of the exceeding the maximum queue length.
Describe the two approaches used to compute Maximum Queue Length;
1) _____________________________________________________________________
________________________________________________________________________
________________________________________________________________________
2) _____________________________________________________________________
________________________________________________________________________
________________________________________________________________________
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STOCHASTIC QUEUING BEHAVIOR
Read 23-4 and 23-6
Page 842
Stochastic delays occur when the demand rate is less than the available capacity (i.e. ρ
____ 1). This is due the _____________________________ in the demand inter-arrival
times and/or service times. These arrival clusters appear due to _________________
and/or ________________________.
As ρ approaches 1 over a long period, the stochastic delays can be come significant.
Stochastic queuing systems are analyzed under _____________________ conditions.
Explain this term _____________________________________________________
___________________________________________________________________
____________________________________________________________________
Explain the term steady state _____________________________________________
_____________________________________________________________________
_____________________________________________________________________
Little’s Law is a description of ______________________________________________
Page 843
Little’s Law
Little’s Law computes the following 4 parameters:
1.
Total Amount of Time Spent by a User in the Queue represented by ______
2.
Total Number of Users in the Queuing System, represented by ___________
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Delay Analysis Workbook
3.
Waiting Time for a User, represented by Wq
4.
Number of Users in the Queue, represented by N
W = sum of amount of time a user spends in the queue and being serviced (Wq).
N = sum of _______________________________________________________
Keep in mind, each of the 4 parameters are random variables. When a queuing system is
in steady-state the expected values (i.e. averages) of the random variables satisfy the
following relationships:
1.
E[N] = λ * E[W]
2.
E[Nq] = = λ * E[Wq]
3.
E[W] = E[Wq] + (1/µ)
Explain each of the relationships above:
1.
__________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
2.
__________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
3.
__________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
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Congestion vs Utilization
Page 843+
Under steady-state conditions, E[W], E[Wq], E[N], and E[Nq] increase non-linearly with
respect to ρ, in proportion to the quantity 1/(1-ρ).
Use a spreadsheet to compute values for 1/(1-ρ) and sketch the relationship ρ on the xaxis) vs 1/(1-ρ) on the y-axis.
ρ
1/(1-ρ).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
The actual vales for E[W], E[Wq], E[N], and E[Nq] depend on the configuration of the
queuing system (i.e. number of servers) and the parameters of the queuing functions. The
most common form of queuing system is an M/G/1 system. This stands for a Memoryless
system with any (general) probability distribution of service times, with 1 server.
Typical M/G/1 system:
•
Single server (e.g. runway)
•
Demand arrives at entirely random times according to a Poisson process
•
Inter-arrival times are described by a negative exponential probability distribution
with parameter λ (i.e. demand rate per unit time)
•
Service rate = µ
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Delay Analysis Workbook
•
Service time S, has variance σ2(S)
•
System has infinite queuing capacity (e.g. taxiway holds all aircraft submitted to a
departure runway queue).
E[Wq] = __________________________________________
E[Wq] = __________________________________________
Now that you have computed E[Wq] you can compute
E[W] = E[Wq] + (1/µ)
Then you can compute,
E[N] = λ * E[W]
E[Nq] = = λ * E[Wq]
Note 1: the expression for E[Wq] includes the term 1/(1-ρ). This term is a dominant
factor in the equations and determines the overall shape of the function.
Note 2: the σ2(S) term, the variability in Service times determines how fast the E[.]
grow. In general, the higher the variability in inter-arrival times (i.e. bunching or
arrivals represented by λ) or the higher the variability in service times (σ2(S)), the
faster E[.] increases.
Exercise #1: Study the relationship between Service Time variability and Average
Number of Users in the Queue.
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Delay Analysis Workbook
Use a spreadsheet to compute and chart the values for ρ vs E[Nq]. Assume an M/G/1
system. See table below. Explain the difference in E[Nq] between system A and B.
What are the design implications of this result?
System A
µ = 60 per hour
(σ2(S) = 0
E[Wq]
E[Nq]
ρ
System B
µ = 60 per hour
(σ2(S) = 0.81
E[Wq]
E[Nq]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Exercise #2: Study the relationship between Demand and Capacity.
Use a spreadsheet to compute and chart the values for ρ vs E[Wq], E[Nq], E[W],
E[N]
Assume an M/G/1 system to model a departure runway during a peak departure
period.
•
Capacity = µ = 48 per hour
•
Service Times: Expected Value = 75 seconds and Standard Deviation = 25
seconds
•
Demand occurs at a steady rate and can be approximated by a Poisson
distribution
Note ρ = Arrival Rate(λ)/Capacity(µ)
System A
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ρ
λ = ρ/µ
E[Wq]
E[Nq]
E[W]
0.6
0.7
0.8
0.9
1
a) Explain what happens as demand approaches capacity?
b) What is the impact on variability as demand approaches capacity?
c) What are the implications for the regulating schedules at airports?
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E[N]
Delay Analysis Workbook
REVIEW:
OVERLOADED QUEUING SYSTEMS
Demand Rate per Unit Time
Service Rate per Unit Time
Time of day
1
2
3
4
5
6
7
8
9
10 11
12 13 14 15
Identify the following parameters. Show all work:
1
Daily Total Capacity of the Arrival Runway (i.e. service rate)
__________________
2
Arrival Rate (i.e. demand rate for period)
_________________________________________
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Delay Analysis Workbook
3
Daily Total Demand for the Arrival Runway
________________________________
4
Time periods with Demand in excess of Capacity
___________________________
5
Time periods in which flights will be delayed
_______________________________
6
Daily Total Demand over Capacity Ratio
_________________________________
7
Do you expect to delays to occur? Why
___________________________________
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Delay Analysis Workbook
________________________________________________________________
______
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Delay Analysis Workbook
Learning Objectives:
1. Understand the dynamics of delays that result from over-scheduled
resources
2. Understand the impact of “exemptions” to the dynamics of delays that
result from over-scheduled resources
3. Understand the impact of “cancellations” to the dynamics of delays that
result from over-scheduled resources
4. Understand the impact of “cancellations with compression” to the
dynamics of delays that result from over-scheduled resources
5. Understand the impact of “slot swapping” to the dynamics of delays that
result from over-scheduled resources
6. Understand the implications of over-scheduled resources on Proportional
Equity
7. Understand the canonical form of the equations for overscheduled
resources
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Delay Analysis Workbook
Baseline Overscheduled Queueing Dynamics
15 flights are scheduled into 8 slots. Each slot can only service one flight at a
time.
1. Complete the Queueing matrix below. Insert each flight into the queue in the
appropriate slot. Use the partial matrix in the middle of the diagram.
CUMULATIVE DETERMINISTIC QUEUEING DIAGRAMS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
F1 F3 F5 F7 F9 F11 F13 F15 F2 F4 F6 F8 F10 F12 F14
Departure Slots
Schedule (AirlineFlight#)
Actual Slot
1 F1
2 F2
F2
3
F3
4
F4
F3
F4
F4
5
F5
F5
6
F6
F6
F6
F6
7
F7
F7
F7
8
F8
F8
F8
F8
F8
F9
F9
F9
F9
9
F5
F7
F9
F10 F10 F10 F10 F10 F10
10
11
F11 F11 F11 F11 F11 F11
12
F12 F12 F12 F12 F12 F12 F12
13
F13 F13 F13 F13 F13 F13 F13
14
F14 F14 F14 F14 F14 F14 F14 F14
Cumulative Number of Departures
F15 F15 F15 F15 F15 F15 F15 F15
FlightDelay (in Slots)
F1
0
F2
1
F3
1
F4
2
F5
2
F6
3
F7
3
F8
4
F9
4
F10
5
F11
5
F12
6
F13
6
F14
7
F15
7
TOTAL 56
Max Users in the Queue =
Total Time Queue Present
Total Users in Queue =
Departure Slot Number
Cumlative Capacity (1 slot per period)
Cumulative Schedule
Takeoff Slot
In Departure Queue
Total Delay Time = Total Flights in Departure Queue
= Area between Cumulative Capacity
and Cumulative Schedule
2. Compute Individual Flight and Total Flight delays. Use the table on the
right hand-side.
3. Identify the Max Users in the Queue
4. Identify the Total Time the Queue is present
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Delay Analysis Workbook
5. Identify the Total Number of Users in the Queue
6. Complete the chart below identifying the number of Flights in the queue
for each slot.
a. Are flights in the non-overscheduled slots affected by
overscheduling in the earlier slots
b. At which slot does the queueing peak
12 # Flights
11
10
Flights in Departure Queue
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Slot #
7. Complete the chart below describing the Delay experienced by each
individual flight.
a. Is flight delay allocated equally for each flight? _________
b. Which flights receive the least delays? ________________
c. Which flights receive the most delays? _________________
d. How much delay does the flight in the non-overscheduled slot
receive? Is ______________
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# Slots Delay for each Flight
11
10
9
8
7
6
5
4
3
2
1
0 F1
F2
F3
F4
F5
F6
F7
F8
F9
F10 F11 F12 F13 F14 F15
Flight Number
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Overscheduled Queueing Dynamics – Exempted Flights
15 flights are scheduled into 8 slots. Each slot can only service one flight at a
time.
Flight F6 is “exempted” from queueing. Exemptions occur for international flights
operating under bi-lateral access agreements, for ”life-guard” flights carrying
donor organs, flights that are not within the scope of the delay allocation rules
(e.g. long distance flights), flights that may have already been allocated (long)
delays.
Exempted flights do not queue. They are given their originally scheduled slots (or
a preferential slot).
1. Complete the Queueing matrix below. Insert each flight into the queue in the
appropriate slot. Use the partial matrix in the middle of the diagram.
1
Departure Slo
Schedule (Air
F1
F2
F1
F2
2
F3
F4
F2
F3
F4
3
F5
F6
F6
F3
F4
F5
4
F7
F8
F3
F4
F5
F7
F8
5
F9
F10
F4
F5
F7
F8
F9
F10
6
F11
F12
F5
F7
F8
F9
F10
F11
F12
Cumulative Number of Departures
7
F13
F14
F7
F8
F9
F10
F11
F12
F13
F14
8
F15
F8
F9
F10
F11
F12
F13
F14
F15
9
-
F9
F10
F11
F12
F13
F14
F15
Max Users in the Queue =
Total Time Queue Present
Total Users in Queue =
Departure Slot Number
Cumlative Capacity (1 slot per period)
Cumulative Schedule
Takeoff Slot
In Departure Queue
Total Delay Time = Total Flights in Departure Queue
= Area between Cumulative Capacity
and Cumulative Schedule
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10
-
F10
F11
F12
F13
F14
F15
11
-
F11
F12
F13
F14
F15
12
-
F12
F13
F14
F15
13
-
14
-
15
-
FlightDelay (in Slots)
F1
0
F2
1
F3
2
F4
3
F5
3
F6
0
F7
3
F8
4
F9
4
F10
5
F11
5
F12
6
F13
F13
6
7
F14 F14
F14
F15 F15 F15
F15
7
TOTAL DELAY
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Delay Analysis Workbook
2. Compute Individual Flight and Total Flight delays. Use the table on the
right hand-side. How do Total Flight Delays compare with the Baseline
queueing? Explain.
3. Identify the Max Users in the Queue. Compare to Baseline queueing?
Explain.
4. Identify the Total Time the Queue is present. Compare to Baseline
queueing? Explain.
5. Identify the Total Number of Users in the Queue. Compare to Baseline
queueing? Explain.
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Delay Analysis Workbook
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Delay Analysis Workbook
Overscheduled Queueing Dynamics – Cancelled Flights
15 flights are scheduled into 8 slots. Each slot can only service one flight at a
time.
Flight F6 is “cancelled”. Cancelled flights occur when the flight is unsafe to
operate (e.g. mechanical failure), or when the airline “tactically” cancels a flight to
save money, a flight-crew is not available, or to re-align aircraft to maintain the
integrity of it’s network.
Cancelled flights are not included in the queue, but their slot remains unused.
1. Complete the Queueing matrix below. Insert each flight into the queue in the
appropriate slot. Use the partial matrix in the middle of the diagram.
1
Departure Slo
Schedule (Air
F1
F2
2
F3
F4
3
F5
F6
4
F7
F8
5
F9
F10
6
F11
F12
7
F13
F14
8
F15
F1
F2
F2
F3
F4
F3
F4
F4
F5
F5
F5
F7
F7
F7
F8
F8
F8
F8
F8
F9
F9
F9
F9
F10
F7
F10
F10
F10
F11
F11
F11
F12
F12
F12
F13
F13
F14
Cumulative Number of Departures
F14
F15
9
-
10
-
11
-
12
-
13
-
14
-
15
-
FlightDelay (in S
F1
0
Departure Slot Number
F2
1
Cumlative Capacity (1 slot per period)
F3
1
Cumulative Schedule
F4
2
Takeoff Slot
F5
2
In Departure Queue
F6
0
F7
3
F8
4
F9
F9
4
F10 F10
F10
5
F11 F11 F11
F11
5
F12 F12 F12 F12
F12
6
F13 F13 F13 F13 F13
F13
6
F14 F14 F14 F14 F14 F14
F14
7
F15 F15 F15 F15 F15 F15 F15
F15
7
TOTAL DELAY
53
Max Users in the Queue =
Total Time Queue Present
Total Users in Queue =
Total Delay Time = Total Flights in Departure Queue
= Area between Cumulative Capacity
and Cumulative Schedule
2. Compute Individual Flight and Total Flight delays. Use the table on the
right hand-side. How do Total Flight Delays compare with the Baseline
queueing? Explain.
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3. Identify the Max Users in the Queue. Compare to Baseline queueing?
Explain.
4. Identify the Total Time the Queue is present. Compare to Baseline
queueing? Explain.
5. Identify the Total Number of Users in the Queue. Compare to Baseline
queueing? Explain.
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Overscheduled Queueing Dynamics – Cancelled Flights with Compression
15 flights are scheduled into 8 slots. Each slot can only service one flight at a
time.
Flight F6 is “cancelled”. This time rather than leave the slot unused, the airline
that operates Flight 6 gives it’s allocated slot away. The subsequent flights are
then “compressed” – each subsequent flight moves up to fill any gaps.
Compression is when unused slots are filled by moving all downstream flights up
to fill the gaps.
Note: Compression can only take place, when the owner of the unused slot
makes this information available.
1. Complete the Queueing matrix below. Insert each flight into the queue in the
appropriate slot. Use the partial matrix in the middle of the diagram.
1
Departure Slots
Schedule (AirlineFlight#)
F1
F2
2
F3
F4
3
F5
F6
4
F7
F8
5
6
7
8
F9
F11 F13 F15 F10 F12 F14
9
10
-
11
-
12
-
13
-
14
-
15
-
Flight Delay (in Slots
Takeoff Slot
F1
F2
F3
F4
F5
In Departure Queue
F6
F12
F12
F13
F13
F13
F13
F14
F14
F14
F14
F14
F14
F15
F15
F15
F15
F15
F15
F7
F8
F9
F10
F11
F12
F13
F14
F15
F1
F2
F2
Departure Slot Number
F3
F3
F4
F4
F4
F5
F5
F5
F7
F7
F7
F8
F8
F8
F8
F9
F9
F9
F9
F10
F10
F10
F10
F10
F11
F11
F11
F11
F11
F12
F12
F12
F12
F13
F13
F14
Cumlative Capacity (1 slot per period)
Cumulative Schedule
Cumulative Number of Departures
F15
TOTAL DELAY
Max Users in the Queue =
Total Time Queue Present
Total Users in Queue =
Total Delay Time = Total Flights in Departure Queue
= Area between Cumulative Capacity
and Cumulative Schedule
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Copyright Lance Sherry 2010
0
1
1
2
2
0
2
3
3
4
4
5
5
6
6
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Delay Analysis Workbook
2. Compute Individual Flight and Total Flight delays. Use the table on the
right hand-side. How do Total Flight Delays compare with the Baseline
queueing? Explain.
3. Identify the Max Users in the Queue. Compare to Baseline queueing?
Explain.
4. Identify the Total Time the Queue is present. Compare to Baseline
queueing? Explain.
5. Identify the Total Number of Users in the Queue. Compare to Baseline
queueing? Explain.
6. Under what conditions would the operator of flight F6 gain by giving up
their unused slot?Explain.
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Overscheduled Queueing Dynamics – Cancelled Flights with Slot Swapping
15 flights are scheduled into 8 slots. Each slot can only service one flight at a
time.
Airline B operates Flights: F2, F4, F6, F8, F10, F12, F14. Flight F6 is cancelled.
F8 is moved in F6 slot, F10 is moved into F8 slot …
The rules for slot swapping: Each airline may freely substitute flights within the
set of it’s own flights slots as long as those flights are not moved earlier than their
originally scheduled slot.
Note: This rule is particularly useful for airlines that cancel flights early in the
over-scheduled period.
1. Complete the Queueing matrix below. Insert each flight into the queue in the
appropriate slot. Use the partial matrix in the middle of the diagram.
1
Departu
Schedu F1
2
F3
F4
F2
3
F5
F6
4
F7
F8
5
F9
F10
6
F11
F12
7
F13
F14
8
F15
9
-
10
-
11
12
-
-
13
-
14
-
15
-
Airline B operates Flights (F2, F4, F6, F8, F10, F12, F14). Flight F6 is cancelled. F8 is moved in F6 slot, F10 is moved into F8 slot …
1
Departu
Schedu F1
F2
2
F3
F4
3
F5
F8
4
F7
F10
5
F9
F12
6
F11
F14
7
F13
8
F15
9
-
10
-
11
12
-
-
13
-
14
-
15
-
Takeoff Slot
Flight Delay (in Slots)
F1
0
F2
1
F3
1
F4
2
F5
2
In Departure Queue
F6
F7
F8
F9
F10
F11
F12
F13
F14
F15
F1
F2
F2
Departure Slot Number
F3
F3
F4
F4
F4
F5
F5
F8
Cumlative Capacity (1 slot per period)
Cumulative Schedule
F5
F8
F8
F8
F7
F7
F7
F7
F10
F10
F10
F10
F10
F9
F9
F9
F9
F12
F12
F12
F12
F12
F12
F11
F11
F11
F11
F11
F11
F14
F14
F14
F14
F14
F14
F14
F13
F13
F13
F13
F13
F13
F13
F15
F15
F15
F15
F15
F15
F9
Cumulative Number of Departures
F15
TOTAL DELAY
Max Users in the Queue =
Total Time Queue Present
Total Users in Queue =
Total Delay Time = Total Flights in Departure Queue
= Area between Cumulative Capacity
and Cumulative Schedule
32
Copyright Lance Sherry 2010
0
3
3
4
4
5
5
6
6
6
48
Delay Analysis Workbook
2. Compute Individual Flight and Total Flight delays. Use the table on the
right hand-side. How do Total Flight Delays compare with the Baseline
queueing? Explain.
3. Identify the Max Users in the Queue. Compare to Baseline queueing?
Explain.
4. Identify the Total Time the Queue is present. Compare to Baseline
queueing? Explain.
5. Identify the Total Number of Users in the Queue. Compare to Baseline
queueing? Explain.
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Copyright Lance Sherry 2010
Delay Analysis Workbook
6. Under what conditions would the operator of flight F6 gain by giving up
their unused slot? Explain.
34
Copyright Lance Sherry 2010
Delay Analysis Workbook
Equity
Every society has rules for sharing goods and burdens among it’s members
(Young,1994). Some resources are managed through property rights and
liabilities that are held and traded by private individuals or held by enterprises
according to complex financial regulations. Other property rights are held by a
governing entity and allocated according to societal needs. The mechanism for
the distribution of property rights expresses the notions of equity in the division of
the resources deemed reasonable by societal norms. The appropriateness of the
equity is determined in part by principle and in part by precedent.
There are three main decisions that must be made in the allocation of commonly
held property: (1) the supply decision determines the amount of resources to be
distributed (e.g. arrival slots). (2) the distributive decision determines the
principles and methods used to allocate the resources (e.g. first-scheduled/firstserved), and (3) the reactive decision: determines the owners or users to the
allocation scheme (e.g. slot substitutions and slot swapping). The focus of this
paper is the implications of the distributive decision.
Varieties of Equity
Equitable allocation should be a simple process. Each party is allocated an equal
distribution measured according to a single yardstick. The reality is that allocated
resources are not equal. Claimant parties are in different situations and the
agreement of single yardstick is difficult to achieve (Rae 1981). A wide range of
philosophers (e.g. Aristotle, Maimonides) have examined the “combinatorics” of
allocation of asymmetric resources to claimants using various yardsticks. These
philosophers have developed appropriate allocation schemes for specific
combinations. One of the emergent themes of these allocation schemes is that
the equity formulas are usually based, either explicitly or implicitly, on a standard
of comparison that ranks the various claimants on their relative desert (Young,
1994; pg 80).
Proportionality and Proportional Equity
One of the oldest and most widely used distributive principles is one that ranks
claimants rights. This is the “Principle of Proportionality.” Proportionality is implicit
in the mechanism of First-Come/First-Serve used in Air Traffic Control (ATC) and
is the explicit in the mechanism of First-Scheduled/First-Served used in Traffic
Flow Management (ATFM). This principle allocates the resources in proportion to
the demand for the resource such that groups (e.g. airlines, passengers with
specific demographics, or flights with specific emissive properties) will receive
delays in proportion to their number of flights scheduled.
Proportional Equity is defined by the following equation:
35
Copyright Lance Sherry 2010
Delay Analysis Workbook
Proportional Equity for Group (i) =
[Total Delay for Group (i) / Total Delay for all Groups] /
[Number of Flights for Group (i) / Total Flights for all Groups]
where i are groups of users (e.g. airlines) 1 through n.
The numerator represents the proportion of delays allocated to Group (i) with
respect to the total delays allocated to all the Groups. The denominator
represents the proportion of flights flown by Group (i).
When the proportion of delays is equivalent to the proportion of flights, the equity
for the group is equal to 1. Proportional equity less that 1, implies that the group
was allocated delays proportionately less than the number of flights scheduled.
This group benefited from the allocation process. Proportional equity greater that
1, implies that the group was allocated delays proportionately more than the
number of flights scheduled. This group was penalized by the allocation process.
Proportional Equity for Individual Flights
80
2.5
2
60
1.5
40
1
20
0.5
0
0
1
8
Proportional Flight
Delay Equity (> 1
Unfair)
Flight Delays (Mins)
The equity of individual flights is entirely based on the magnitude of the spill-over
of preceding flights. As a consequence, flights scheduled early in the congested
period receive less that the average delay and have a proportional equity of less
than 1. Flights scheduled late in the congested period, receive more delays and
experience an equity of greater than 1. These results are illustrated in Figure 5.
The vertical bars represent the delays allocated to individual flights (left y-axis).
The magenta line defines the proportional equity for each flight (right y-axis).
15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190 197
Flights by Seqeuence of Schedule
Proportional equity for individual flights is asymmetric. Flights are shown from
first scheduled to last scheduled. The left y-axis identifies the delays experienced
by each flight. The right y-axis identifies the proportional equity of each flight.
Flights with proportional equity less than one experience an allocation advantage.
Flights with proportional equity greater than one experience an allocation
disadvantage.
Figure 5.
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Copyright Lance Sherry 2010
Delay Analysis Workbook
Proportional Equity for Groups of Flights
Flights are scheduled by the individual airlines to meet the airlines network and
connecting passenger and crew objectives and are constrained by the available
airline resources. When flights are scheduled by independent airlines, without a
priori knowledge of the schedules of other airlines, over-scheduling is feasible.
Further, the independent scheduling results in an aggregate schedule that does
not follow any rigid structure that could lead to symmetry in allocation of flight
delays or equity. This schedule is effectively generated by a random process.
Groupings of scheduled flights will experience proportional equity relative to the
position of the flights in the schedule. A group of flights disproportionately
positioned in the front (or back) of the congested period will experience less (or
more) proportional equity.
Exercise:
Assume Flights are operated by airlines as follow:
Airline A operates Flights: F1, F7, F9, F11, F13, F15
Airline B operates Flights: F2, F4, F6, F8, F10, F12, F14.
Airline C operates Flights: F3, F5
Compute the Proportional Equity for each Airline and
Baselin
e
Exemptio
n
Total
Delays
Total
Number of
Flights
Airlin
eA
Airlin
eB
Total
Delays
Number of
Flights
Proportion
al Equity
Total
Delays
Number of
Flights
Proportion
37
Copyright Lance Sherry 2010
Cancelle
d
Cancelled
with
Compressio
n
Cancelle
d with
Swappin
g
Delay Analysis Workbook
Airlin
eC
al Equity
Total
Delays
Number of
Flights
Proportion
al Equity
1. Under which queueing scenario is Inter-airline Proportional Equity best (i.e.
Airline Proportional Equity closest to 1)? Explain.
2. What role does position in the schedule have on Inter-airline Proportional
Equity ? Explain.
3. How does Cancellations, Cancellations with Compression, and Slot
Swapping affect Inter-airline Proportional Equity? Explain.
38
Copyright Lance Sherry 2010
Delay Analysis Workbook
39
Copyright Lance Sherry 2010
Delay Analysis Workbook
GENERALIZED MODEL OF OVERSCHEDULED QUEUEING DELAYS
A canonical representation of the scheduled over-utilization of a scarce resource
is illustrated in Figure 1. In this scenario, flights are scheduled in 15 minutes
periods. Starting at 15 minute period #6, flights are scheduled in excess of the
available capacity for a period known as the congested period. Following the
congested period, the low-demand provides a reservoir to absorb spill-over
delays. The flights are numbered in the order in which they are scheduled.
Flight Sequence (Bottom-up) per 15 Minute Time Slot
High-Demand
15
30
45
60
75
90
105
120
135
150
14
28
44
59
74
89
104
119
134
149
13
28
43
58
73
88
103
118
133
148
12
27
42
57
72
87
102
117
132
147
11
26
41
56
71
86
101
116
131
146
10
25
40
55
70
85
100
115
130
145
9
24
39
54
69
84
99
114
129
144
8
23
38
53
68
83
98
113
128
143
7
22
37
52
67
82
97
112
127
142
6
21
36
51
66
81
96
111
126
141
XXX XXX XXX XXX XXX
5
20
35
50
65
80
95
110
125
140
155
160
165
170
175
180
185
190
195
200
XXX XXX XXX XXX XXX
4
19
34
49
64
79
94
109
124
139
154
159
164
169
174
179
184
189
194
199
XXX XXX XXX XXX XXX
3
18
33
48
63
78
93
108
123
138
153
158
163
168
173
178
183
188
193
198
XXX XXX XXX XXX XXX
2
17
32
47
62
77
92
107
122
137
152
157
162
167
172
177
182
187
192
197
XXX XXX XXX XXX XXX
1
16
31
46
61
76
91
106
121
136
151
156
161
166
171
176
181
186
191
196
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Arrival Capacity
1
2
3
4
5
6
Low-Demand
15 minute Time Slots
Congested Period
Scheduled over-utilization of a scarce resource.
Figure 1
The systemic allocation of delays to individual flights according to the order of
scheduling is illustrated in Figure 2. The color-code identifies the degree of delay
experienced by each flight. Flights in excess of the capacity in the first 15 minute
period, spill-over into the second 15 minute period escalating the delays
assigned to these flights. The spill-over cascades through the congested period
(15 minute time slots 6 through 15) and into the period following the congested
period (15 minute time slots 16 through 25).
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Copyright Lance Sherry 2010
Delay Analysis Workbook
Legend
No Delay
15 Min Dely
30 Min Delay
45 Min Delay
60 Min Delay
75 Min Delay
Flight Sequence (Bottom-up) per 15 Minute Time Slot
Arrival Capacity
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
9
19
29
39
49
59
69
79
89
99
109
119
129
139
149
159
169
179
189
199
8
18
28
38
48
58
68
78
88
98
108
118
128
138
148
158
168
178
188
198
7
17
27
37
47
57
67
77
87
97
107
117
127
137
147
157
167
177
187
197
6
16
26
36
46
56
66
76
86
96
106
116
126
136
146
156
166
176
186
196
XXX XXX XXX XXX XXX
5
15
25
35
45
55
65
75
85
95
105
115
125
135
145
155
165
175
185
195
XXX XXX XXX XXX XXX
4
14
24
34
44
54
64
74
84
94
104
114
124
134
144
154
164
174
184
194
XXX XXX XXX XXX XXX
3
13
23
33
43
53
63
73
83
93
103
113
123
133
143
153
163
173
183
193
XXX XXX XXX XXX XXX
2
12
22
32
42
52
62
72
82
92
102
112
122
132
142
152
162
172
182
192
XXX XXX XXX XXX XXX
1
11
21
31
41
51
61
71
81
91
101
111
121
131
141
151
161
171
181
191
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
1
2
3
4
5
15 minute Time Slot
Flights allocated slots within the available capacity. Note that the effect of
scheduled over-utilization spills-over resulting in increasing delays to flights later
in the schedule. Delays also spill-over into period after the congested period.
Figure 2.
The sum of the individual flight delays, known as the Total Delay Time, is a
function of the degree of over-scheduling as well as the reservoir of capacity
following the congested period. As shown in Figure 3, a queue of flights builds
until the end of the congested period, at which time the reservoir of capacity
allows the queue to dissipate.
# of
Operations
NOT DRAWN TO SCALE
DemandHigh (Ops/15 mins)
Capacity (Ops/15 mins)
DemandLow (Ops/15 mins)
Time-of-Day (15 min bins)
Max # of Users in Queue
Time OverScheduled
Total Time with Queue Present
Queueing of flights due to over-scheduling. The Total Delay time is the area
under the Queueing function.
Figure 3
41
Copyright Lance Sherry 2010
Delay Analysis Workbook
The Total Delay Time resulting from the scheduled utilization is defined by the
equation:
Total Delay Time =
½ * (Duration of Congested Period)2* (HighDemand – Capacity) *
[1 + HighDemand – Capacity ]
Capacity – LowDemand
This equation is derived by calculating the area under the queue curve in Figure
3. The equation highlights the following properties:
(1)
Conservation of Total Delays. The Total Delay is independent of the
order of the flights. The Total Delay is dependent on the relationship
amongst the four terms: Capacity, High_Demand, Low_Demand and
Conegsted_Period. Saying this another way, the only way to reduce
the Total Delay is to remove flights.
Duration of Congested Period is Critical The factor in the equation that
has the biggest effect is the duration of the congested period
(TCongestedPeriod). This term is squared. For every additional unit of time
in the congested period, the Total Delays increase geometrically.
Reservoirs are Critical: The Total Delays is not only dependent on the
degree of over-scheduling, but also on the degree of under-scheduling
after the congested period. The degree of under-scheduling provides a
reservoir to absorb the spill-over from the congested period. A low
degree of under-scheduling can result in extending the queue
significantly.
(2)
(3)
Flight Delays (Mins)
The delays experienced by individual flights are shown in Figure 4. Flights early
in the congested period experience relatively low delays. Flights at the end of the
congested period and flights right after the congested period experience the
highest delays.
80
60
40
20
0
1
8
15
22
29
36
43
50
57
64
71
78
85
92
99 106 113 120 127 134 141 148 155 162 169 176 183 190 197
Flights by Seqeuence of Schedule
Delays experienced by individual flights
Figure 4.
42
Copyright Lance Sherry 2010
Delay Analysis Workbook
(4)
Asymmettry of Individual Flight Delays: The delays assigned to
individual flights are a function of the location of the flight in the
schedule. Flights scheduled early in the congested period, are
allocated less delays than those flights later in the congested period.
Max Users in Queue =
(DemandHigh – Capacity) * TOverScheduled
Determined by:
• Degree of Over-scheduling
• Duration of Over-scheduling
Total Time Queue Present =
TOverScheduled + (DemandHigh – Capacity) * TOverScheduled
(Capacity – DemandLow)
Spill-over effect depends on degree of over scheduling and available capacity
after the over-scheduled slots
Total Users in Queue =
//Users in the Overscheduled region minus the 1st batch that do not queue
[ TOverScheduled * (DemandHigh) ] – [1 * Capacity] +
//Users in the duration of the queue
[ (DemandHigh – Capacity) * TOverScheduled ] * DemandLow
(Capacity – DemandLow)
Exercise:
43
Copyright Lance Sherry 2010
Delay Analysis Workbook
60
16
14
12
10
8
6
4
2
0
50
40
30
20
10
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Time of Day (15 min Bins)
Demand
Capacity
Queue Length
Capacity = 10 flights in 15 minute period
HighDemand = 15 flights in 15 minute period
LowDemand = 5 flights in 15 minute period
Congested Period = T = 10 15 minute periods.
Compute the following measures. Show all work.
1. Max Users in the Queue
2. Total Time Queue Present
3. Total Delay Time
4. Total Users in the Queue
5. Expected Delay for Users in the Queue
Answer Key:
44
Copyright Lance Sherry 2010
Queue Length
No. of Users
Demand v. Capacity Bar Chart (with Que Length)
Copyright Lance Sherry 2010
45
10 Capacity
5 Demand Low
15 Demand High
10 Congested Period
Max Users in Queue
50
Total Time with Queue Present
20
Total Delay Time
500
Total Users in the Queue
190
Expected Delay for Users in Queue
2.63157895
Delay Analysis Workbook