ICRAT Budapest Hungary June 2010 ThroughputComplexity Tradeoffs for
ICRAT Budapest, Hungary June, 2010 Throughput/Complexity Tradeoffs for Routing Traffic in the Presence of Dynamic Weather Presented by: Valentin Polishchuk, Ph. D. June, 2010
Team of Collaborators • Jimmy Krozel, Ph. D. , Metron Aviation, Inc. , USA • Joseph S. B. Mitchell, Ph. D. , Applied Math, Stony Brook University, USA • Valentin Polishchuk, Ph. D. , and Anne Pääkkö, Computer Science, University of Helsinki, Finland Funding provided by: Academy of Finland, NASA and NSF ICRAT ’ 10 Budapest, Hungary June, 2010
Algorithmic Problem • Given weather-impacted airspace • Find weather-avoiding trajectories for aircraft • Assumptions en-route fixed flight level (2 D, xy) generally unidirectional (e. g. , East-to-West) flow ICRAT ’ 10 Budapest, Hungary June, 2010
Airspace Sector ICRAT ’ 10 Budapest, Hungary June, 2010
Airspace Center ICRAT ’ 10 Budapest, Hungary June, 2010
FC A Airspace FCA ICRAT ’ 10 Budapest, Hungary June, 2010
Generic Model • Polygonal domain – outer boundary • source and sink edges – obstacles Sink Source • weather, no-fly zones ICRAT ’ 10 Budapest, Hungary June, 2010
Aircraft: Disk • Radius = RNP = 5 nmi ICRAT ’ 10 Budapest, Hungary June, 2010
Airlane: “thick path” • Thickness = 2*RNP = 10 nmi MIT = 10 nmi ICRAT ’ 10 Budapest, Hungary June, 2010
Algorithmic Problem • Given weather-impacted airspace • Find weather-avoiding trajectories for aircraft ICRAT ’ 10 Budapest, Hungary June, 2010
Model • Given polygonal domain with obstacles, source and sink • Find thick paths pairwise-disjoint avoiding obstacles ICRAT ’ 10 Budapest, Hungary June, 2010
Solution: Search Underlying Grid ICRAT ’ 10 Budapest, Hungary June, 2010
Hexagonal disk packing in free space ICRAT ’ 10 Budapest, Hungary June, 2010
Graph • Nodes: disks • Edges between touching disks • Source, sink • Every node has capacity 1 ICRAT ’ 10 Budapest, Hungary June, 2010
Source-Sink Flow • Decomposes into disjoint paths ICRAT ’ 10 Budapest, Hungary June, 2010
Source-Sink Flow • Decomposes into disjoint paths Max. Flow → Max # of paths Min. Cost Flow → Shortest paths • Inflate the paths ICRAT ’ 10 Budapest, Hungary June, 2010
Examples ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
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Additional constraints: Sector boundaries crossing Communication between ATCs ICRAT ’ 10 Budapest, Hungary June, 2010
Higher cost for crossing edges in the graph ICRAT ’ 10 Budapest, Hungary June, 2010
Conforming flow ICRAT ’ 10 Budapest, Hungary June, 2010
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Theoretical guarantee: Max # of paths Maximum Flow Rates for Capacity Estimation in Level Flight with Convective Weather Constraints Krozel, Mitchell, P, Prete Air Traffic Control Quarterly 15(3): 209 -238, 2007 Capacity = length of shortest B-T path in “critical graph” ℓij = floor(dij/w) ICRAT ’ 10 Budapest, Hungary June, 2010
Moving obstacles? • Paths become infeasible ICRAT ’ 10 Budapest, Hungary June, 2010
Free. Flight ICRAT ’ 10 Budapest, Hungary June, 2010
Solution: Search Time-Expanded Grid ICRAT ’ 10 Budapest, Hungary June, 2010
Lifting to (x, y, t) ICRAT ’ 10 Budapest, Hungary June, 2010
Obstacles ICRAT ’ 10 Budapest, Hungary June, 2010
Time Slicing ICRAT ’ 10 Budapest, Hungary June, 2010
Disk Packings ICRAT ’ 10 Budapest, Hungary June, 2010
Edges ICRAT ’ 10 Budapest, Hungary June, 2010
Node Capacity = 1 ICRAT ’ 10 Budapest, Hungary June, 2010
Supersource, supersink ICRAT ’ 10 Budapest, Hungary June, 2010
Supersource-supersink flow ICRAT ’ 10 Budapest, Hungary June, 2010
Examples ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
Holding ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
Holding ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
The two extremes • Static airlanes – coherent traffic – not adjustable to dynamic constraints • Flexible flow corridors – paths, morphing with obstacles motion – keep threading amidst obstacles • Free. Flight – fully dynamic – “ATC nightmare” ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
Computing the Corridors • Decide – how many are possible – threading amidst obstacles • At every time slice – route paths – with given threadings – Shortest paths • • “pulled taut” against obstacles → morph slowly ICRAT ’ 10 Budapest, Hungary June, 2010
Experiments ICRAT ’ 10 Budapest, Hungary June, 2010
Airspace • 300 x 210 nmi rectangle • Weather Severity Index (WSI) – percentage of space covered with obstacles Weather organizations – Popcorn Convection (PC) • scattered obstacles – Squall Line (SL) • aligned obstacles ICRAT ’ 10 Budapest, Hungary June, 2010
Setup • For WSI = 0, 10, …, 60 – until reaching WSI • generate random obstacle • place it randomly in the airspace • Random velocity • Squall Line – WSI = 0, 5, …, 35 ICRAT ’ 10 Budapest, Hungary June, 2010
100 instances for each WSI Compute trajectories • Static • Free. Flight speed = 420 knots ICRAT ’ 10 Budapest, Hungary June, 2010 • Corridors
Traffic Complexity • Average over time and tiles • In a tile, at a time – # of aircraft – Var(velocites) ICRAT ’ 10 Budapest, Hungary June, 2010
Complexity (100 instances / WSI) ICRAT ’ 10 Budapest, Hungary June, 2010
Throughput (100 instances / WSI), aircraft /. 5 hr ICRAT ’ 10 Budapest, Hungary June, 2010
Summary • Airspace capacity estimation Fundamental research question: can study either theoretically or empirically At the root of Traffic Flow Management (TFM): How do you know that you have a TFM problem, Demand > Airspace Capacity, unless you have a good way of estimating the airspace capacity? Capacity ≠ function( airspace ) • Different paradigms → different capacity → different complexity • Operational requirements – e. g. , conforming flows • Temporal component – e. g. , holding Help in quantifying tradeoffs ICRAT ’ 10 Budapest, Hungary June, 2010
Future Research • Sensitivity to complexity parameters • Route Planning in Terminal or Transition Airspace – Trees (e. g. , STARS) • static • “free”? • flexible • Further Dimensions – Multiple Altitudes, Directions of Flows – 4 D Space-Time Constraints (flow and weather constraints) – Different route types • Real Weather ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
ICRAT ’ 10 Budapest, Hungary June, 2010
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