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Airport Taxi Operations Modeling: GreenSim John Shortle, Rajesh Ganesan, Liya Wang, Lance Sherry,Terry Thompson, C.H. Chen September 28, 2007 CENTER FOR AIR TRANSPORTATION SYSTEMS RESEARCH Outline CATSR • Queueing 101 • GreenSim: Modeling and Analysis • Tool Demonstration & Case Study 2 Motivation CATSR • GreenSim • Airport as a “black-box” – 5 stage queueing model • Schedule and configuration dependent – Queueing models configured based on historic data • Stochastic behavior (i.e. Monte Carlo) – Behavior determined by distributions • Comparison of procedures (e.g. RNP procedures) and technologies (e.g. surface management) – Adjust distributions to reflect changes • Rapid (< 1 week) – Fast set-up and run Typical Queueing Process CATSR Customers Arrive Common Notation l: Arrival Rate (e.g., customer arrivals per hour) m: Service Rate (e.g., service completions per hour) 1/m: Expected time to complete service for one customer r: Utilization: r = l / m A Simple Deterministic Queue CATSR l m • Customers arrive at 1 min, 2 min, 3 min, etc. • Service times are exactly 1 minute. • What happens? Customers in system 1 2 3 4 5 6 7 Arrival Time (min) Departure A Stochastic Queue CATSR • Times between arrivals are ½ min. or 1½ min. (50% each) • Service times are ½ min. or 1½ min. (50% each) • Average inter-arrival time = 1 minute • Average service time = 1 minute • What happens? Arrival Departure Customers in system 1 2 3 4 5 6 7 Time (min) Service Times Stochastic Queue in the Limit CATSR 100 90 80 Wait in 70 60 Queue 50 (min) 40 30 20 10 0 0 500 1000 1500 2000 2500 3000 3500 4000 Customer # • Two queues with same average arrival and service rates • Deterministic queue: zero wait in queue for every customer • Stochastic queue: wait in queue grows without bound 7 • Variance is an enemy of queueing systems The M/M/1 Queue CATSR 0.018 0.016 Gamma A single server 0.014 0.012 Normal 0.01 f(x) Inter-arrival times Service times follow an 0.008 0.006 0.004 follow an exponential distribution 0.002 0 Exponential exponential 50 40 60 80 100 120 140 x 160 180 200 220 240 distribution 45 (or arrival process 40 is Poisson) 35 30 L 25 20 r 15 L= 10 1 r 5 0 0 0.2 0.4 0.6 0.8 1 1.2 Avg. # in r System The M/M/1 Queue CATSR • Observations – 100% utilization is not desired • Limitations – Model assumes steady-state. Solution does not exist when r > 1 (arrival rate exceed service rate). – Poisson arrivals can be a reasonable assumption – Exponential service distribution is usually a bad assumption. 50 45 40 r 35 L= 30 L 25 1 r 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1 1.2 r The M/G/1 Queue CATSR Service times follow a general distribution 35 Required inputs: 30 • l: arrival rate 25 L • 1/m: expected service time 20 • : 15 std. dev. of service time 10 5 0 0 0.2 0.4 0.6 0.8 1 1.2 r m = 1, = 0.5 r l 2 2 2 L=r 2(1 r ) Avg. # in System M/G/1: Effect of Variance CATSR 18 16 14 12 L 10 Exponential 8 Service Deterministic 6 Arrival Rate Service Service Rate 4 Held 2 Constant 0 0 0.5 1 1.5 2 2.5 3 r 2 l2 2 L=r l = 0.8, m = 1 (r = 0.8) 2(1 r ) Other Queues CATSR • G/G/1 – No simple analytical formulas – Approximations exist • G/G/∞ – Infinite number of servers – no wait in queue – Time in system = time in service • M(t)/M(t)/1 – Arrival rate and service rate vary in time – Arrival rate can be temporarily bigger than service 12 rate Queueing Theory Summary CATSR • Strengths – Demonstrates basic relationships between delay and statistical properties of arrival and service processes – Quantifies cost of variability in the process – Analytical models easy to compute • Potential abuses – Only simple models are analytically tractable – Analytical formulas generally assume steady-state – Theoretical models can predict exceptionally high delays – Correlation in arrival process often ignored • Simulation can be used to overcome limitations 13 GreenSim CATSR 14 GreenSim Input/Output Model CATSR Outputs: User adjustable input: · Delays (taxi-in, taxi-out) · Flights Schedule GreenSim Airport · ADOC · Aircraft type Operations · Emission · Capacity (AAR, ADR) Simulation · Departure Runway Queue Size · Gate Utilization 15 GreenSim Architecture CATSR Airport Performance Data Analysis Simulation Analysis Error Actual taxi-in and report Simulation taxi-in taxi-out times and taxi-out times - ASPM Databases Airport Service Times Settings Data performance Individual enhancement processing Taxi-in, taxi-out model AAR, ADR times Airport Capacity Settings Airport Data simulation Airport processing Settings Queueing model Environmental analysis Arrival and Departure External Emission model Schedules demands Database 16 Data Analysis Process CATSR 17 Queueing Simulation Model CATSR NAS lA G / G / / / FCFS Runway Runway G / G / 1 / / FCFS m m DR AR Taxiway G / G / / / FCFS Taxi-out G / G / / / FCFS Taxiway Taxi-in times m DT m AT times Ready to lD Turn depart around reservoir 18 Service Times Settings CATSR Segment Name Settings Notation Arrival Runway S1~Exponential(1/AAR) Arrival Taxiway S2~NOMTI + DLATI- S1 DLATI~Normal (u1 , 1 ) Departure S3~NOMTO + DLATO- S4 DLATO~Normal (u 2 , 2 ) Taxiway Departure S4~Exponential(1/ADR) Runway 19 Performance Analysis CATSR • Delays (individual, quarterly average, hourly average, daily average) • Fuel Fuel = (TIM j ) ( FFj /1000) ( NE j ) j • Emission (HC, CO, NOx, SOx) Emissioni = (TIM j ) ( FFj /1000) ( NE j ) EI ij j TIMj = taxi time for type-j aircraft FFj = fuel flow per time per engine for type-j aircraft NEj = number of engines used for type-j aircraft EIij = emissions of pollutant i per unit fuel consumed 20 for type-j aircraft EWR Hourly Average Delays CATSR 21 CATSR EWR Quarterly-Hour Average Delays CATSR 23 EWR Daily Average Delays CATSR 24 CATSR You cannot always replace a random variable with its average value. 25 The M/M/1 Queue CATSR A single server Service times follow an Inter-arrival times exponential distribution follow an 0.018 exponential 0.016 Gamma distribution 0.014 (or arrival process 0.012 is Poisson) Normal 0.01 f(x) 0.008 0.006 0.004 0.002 Exponential 0 40 60 80 100 120 140 160 180 200 220 240 x Poisson Process CATSR • For many queueing systems, the arrival process is assumed to be a Poisson process. • There are good reasons for assuming a Poisson process – Roughly, the superposition of a large number of independent (and stationary) processes is a Poisson process. • However, the assumption is over-used. • For a Poisson process, inter-arrival times follow an exponential distribution. 27 Notation for Queues CATSR A/B/C/D/E M for Markovian A=interarrival time distribution (exponential) distribution B=service time distribution G for General C= Number of servers (arbitrary) distribution D=Queueing Size Limit . E=Service Discipline(FCFS, LCFS, Priority, etc.) Examples M/M/1 M/G/1 M/G/1/K M/G/1/Infinity/Priority Poisson Distribution CATSR • Ladislaus Bortkiewicz, 1898 • Data – 10 Prussian army corps units observed over 20 years (200 data points) – A count of men killed by a horse kick each year by unit – Total observed deaths: 122 • Number of deaths (per unit per year) is a Poisson RV with mean 122 / 200 = 0.61. 120 100 Number Theoretical Observed 0 108.67 109 Occurrences 80 1 66.29 65 Theoretical 2 20.22 22 60 Observed 3 4.11 3 4 0.63 1 40 5 0.08 0 6 0.01 0 20 0 0 1 2 3 4 5 6 Deaths per Unit per Year Performance Analysis CATSR • Delays (individual, quarterly average, hourly average, daily average) • Fuel Fuel = (TIM ) (FF /1000) ( NE ) j j j j • Emission (HC, CO, NOx, SOx) Emissioni = (TIM j ) ( FFj /1000) ( NE j ) EI ij j • Cost (Directing Operation Cost and Delay Cost) Total DOC = Fuel Cost Operationa l Cost = Fuel Cost Taxi - in Operation Cost Taxi - out Operation Cost = Fuel Fuel Price (Taxi - in Taxi - out times) DOCPrice 30 Where DOCPrice = (Labor Maintenanc e Owership Other) = 38.24$/min Some Common Distributions CATSR Form of Probability 0.018 Density Function (PDF) 0.016 Gamma Normal 1 e ( x m ) /(2 ) 0.014 f ( x) = 2 2 0.012 2 Normal 0.01 Gamma f(x) 1 0.008 f ( x) = x 1e x / ( ) 0.006 0.004 Exponential f ( x) = l e l x 0.002 Exponential 0 40 60 80 100 120 140 160 180 200 220 240 x Mean = 106 Std. Dev. = 26.9 (normal and gamma) Some Common Distributions CATSR Form of Probability Density Function (PDF) Normal 1 f ( x) = e ( x m ) /(2 ) 2 2 0.018 0.016 Gamma 2 0.014 0.012 Normal 0.01 Gamma f(x) 0.008 1 0.006 f ( x) = x 1e x / 0.004 ( ) 0.002 0 Exponential 40 60 80 100 120 140 160 180 200 220 240 x Exponential f ( x) = l e l x Mean = 106 Std. Dev. = 26.9 (normal and gamma) Queueing Theory CATSR Queueing Theory: The theoretical study of waiting lines, expressed in mathematical terms • How long does a customer wait in line? • How many customers are typically waiting in line? • What is the probability of waiting longer than x seconds in line? 33 Queueing 101 CATSR Queue Teller Customers Need vs. Capability CATSR • Mismatch between demand for analysis and capability of existing tools • Analysis Capability: • Airport as Single Queue (Odoni) – Not enough resolution • Airport as discrete event simulation (TAAM, Simmod) – Too much resolution requires: • Time-consuming detailed rules for individual flight behavior • Long run times The M/G/1 Queue CATSR Service times follow a general distribution 35 Required inputs: 30 • l: arrival rate 25 L • 1/m: expected service time 20 • : 15 std. dev. of service time 10 5 0 0 0.2 0.4 0.6 0.8 1 1.2 r m = 1, = 0.5 r l 2 2 2 L=r 2(1 r ) Avg. # in System M/G/1: Effect of Variance CATSR 18 16 14 12 L 10 Exponential 8 Service Deterministic 6 Arrival Rate Service Service Rate 4 Held 2 Constant 0 0 0.5 1 1.5 2 2.5 3 r 2 l2 2 L=r l = 0.8, m = 1 (r = 0.8) 2(1 r )