Scheduling Algorithms for Automated Traffic

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Scheduling Algorithms for Automated Traffic

• Arvind Giridhar
• Joint work with Prof. P.R. Kumar

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I.T. Convergence Lab
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Control of Automated Traffic

• Must efficiently transport vehicles from
  specified origins to destinations, on a given
  network of roads.

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Control Hierarchy
Path planning layer
Timed Trajectories
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Task of Path Planner

• Given (starting) locations and final destinations
  of finite set of cars on a road network.
• Must provide timed trajectories to all cars so
  that
• Minimum separation in between cars is maintained.
  
• System is cleared, i.e., no deadlock.
• Cars exit system on reaching destinations.

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Discretizing the System
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Discretizing the System

• Split lanes into sections.
• Section width equal to minimum safe distance.
• Only one car per section.

• Slotted time
• All cars have same fixed velocity, or are
  stationary.
• Cars can move from one section to adjacent
  section in a single time slot.

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Graph Representation

• Directed graph.
• Each node represents a section of a lane.
• Cars occupy nodes at most one car per node.
• Car can move to adjacent node in 1 time slot.
• Edge conflicts disallow simultaneous movements.

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Rules for Movements
Platoon of cars can shift simultaneously.
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Rules for Movements
Cycle of cars can be shifted simultaneously
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Rules for Movements
Cars cannot be shifted across conflicting edges
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Output of Path Planner

• Timed trajectory
• Spatial component A directed walk on the graph
  from source to destination.
• Time component In each time slot
• Car either remains in current node (stop), or
• Shifts across edge to the next node in
  route(go).
• Trajectories must follow graph constraints.

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Architectural separation

• Possible approach Separate scheduling and
  routing.

Router
Scheduler
Suboptimal, but simplifies design of control
system
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Implications of Separation

• Routing decisions can be made independently
• Scheduler must provide guarantees
• Must be able to feasibly schedule arbitrary (or
  at least large class) of routes
• .with no collisions or deadlocks.
• In other words, clear the system while
  maintaining graph constraints

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Task of Scheduler

• Given routes on graph for each car.
• Output schedule for each car sequence of binary
  instructions for each time slot
• Stop Stay in current node.
• Go Go to next node in corresponding route.

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Graph Scheduling Problem

• Given routes, can the cars be feasibly scheduled?
• If so, provide schedules for each car so that
• All cars reach their destinations along
  respective routes.
• No collisions or deadlock.
• Also, schedules must be efficient, i.e. clear
  the system as quickly as possible.

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Definitions

• Occupied cycle directed cycle of occupied nodes
  and shift edges.
• Occupied path
• Directed path of occupied nodes and shift edges.
• Terminating node is either unoccupied or belongs
  to occupied cycle.

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Feasibility of Scheduling

• Deadlock a configuration in which some subset
  of cars cannot be shifted.

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Occupied cycle leading to deadlock
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Deadlock!
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Sufficient Condition for Schedulability

• Condition every vertex in the graph has either
  in-degree or out-degree (or both) equal to one.
• Theorem if the initial configuration contains no
  occupied cycles, then there exists a feasible
  schedule that clears the system.
• Note Similar result derived by Fanti (97),
  Lawley (01) for Flexible Manufacturing Systems.

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Proof

• Suppose no occupied cycle at time t.
• Enough to prove a single car can be shifted, such
  that
• Resulting configuration has no occupied cycle.
• Consider any occupied path terminating in
  unoccupied node. .

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Case 1 Indegree 1
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Case 2 Out degree 1
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Case 2.5 Out-degree 1
So no occupied cycles created.
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Validity of degree condition

• Assumption satisfied by all road networks
  consisting of two lane roads and intersections.
• In multi-lane roads, could satisfy assumption by
  restricting lane changes (e.g. to every alternate
  section).

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Implications of Result

• Existence of feasible schedule depends only on
  current state and next step in routes.
• Independent of future routes.
• Allows recalculation of schedules on the fly.
• Allows design of myopic algorithms.

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Optimization of Performance

• Cost criterion Time to clear the system.
• Optimal schedule given initial configuration,
  the schedule that clears the system in minimum
  time.
• Theorem Finding the optimal schedule for an
  arbitrary graph with arbitrary initial
  configuration is NP-complete.

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One-step Moves

• Given a configuration and next step of all
  routes, maximize the number of cars moved in a
  single time slot.
• Maximum feasible subset problem Find the
  largest feasible subset of cars.
• Feasible subset subset that can be
  simultaneously moved, resulting in configuration
  having no occupied cycle.
• This problem is also NP-complete!

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Polynomial Time Suboptimal Algorithm

• Descent-like algorithm.
• Starts from some maximal feasible set.
• Searches among neighbors for larger set
• for a suitably defined notion of neighborhood.
• Stops when a set is larger than all its
  neighbors.
• Polynomial time.
• Guaranteed to clear the system.

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Implementation Issues

• Schedule can be calculated a-priori, or at each
  step.
• Algorithms can be state dependent
• Only current configuration and future routes
  required.
• Some initial assumptions can be relaxed
• Cars can enter the system.
• Routes can be changed online.
• as long as no occupied cycles result!

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Implementation Issues

• Control of the system need be exercised only at
  traffic lights.
• Decisions only need to be made at intersection
  nodes .
• Cars occupying every other node shift if the node
  ahead is available in the next time slot.

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Critique of Model

• Effect of discretization
• Imposes an unrealistic level of homogeneity, e.g.
  all nodes, i.e. sections of roads are equal, all
  speeds are equal
• However, result shows that a large class of
  routes can be feasibly scheduled within such a
  restricted class of schedules

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Critique of Model

• Imposes binary behavior (stop or go from slot
  to slot) on a continuous system
• Cellular automaton models in transportation
  literature successfully capture behavior of
  congested traffic
• Such a model is more suitable for providing
  guarantees
• Scheduler provides guarantees in terms of
  discrete deterministic model
• Model itself is designed to accept tolerances of
  individual car controllers

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Conclusions

• Have provided zeroeth order model and class of
  schedules.
• Satisfactory for controlling remote controlled
  cars!
• Result shows existence of large class of deadlock
  free schedules for (nearly) arbitrary routes.
• More sophisticated discrete models could be
  suitable for actual automated traffic system.