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Queues and Network Control for Urban Traffic Systems Workshop on Control for Networked Transportation Systems July 8 2019 Ketan Savla ( ksavla@usc.edu ) University of Southern California Ketan Savla (USC) CNTS Workshop July 8 2019 1 / 11

  1. Queues and Network Control for Urban Traffic Systems Workshop on Control for Networked Transportation Systems July 8 2019 Ketan Savla ( ksavla@usc.edu ) University of Southern California Ketan Savla (USC) CNTS Workshop July 8 2019 1 / 11

  2. Queues and Network Control for Urban Traffic Systems Workshop on Control for Networked Transportation Systems July 8 2019 Ketan Savla ( ksavla@usc.edu ) University of Southern California Thanks to NSF EPCN and DCSD, CALTRANS Ketan Savla (USC) CNTS Workshop July 8 2019 1 / 11

  3. Overview Transportation Science Operations Research Control Theory Ketan Savla (USC) CNTS Workshop July 8 2019 2 / 11

  4. Overview Transportation Science Queues Network Control Operations Research Control Theory Symbiosis between transportation and systems sciences Ketan Savla (USC) CNTS Workshop July 8 2019 2 / 11

  5. Overview DATA Transportation Science Queues Network Control Operations Research Control Theory Symbiosis between transportation and systems sciences Tight integration essential for efficient use of data Ketan Savla (USC) CNTS Workshop July 8 2019 2 / 11

  6. Transportation Queues jobs server Ketan Savla (USC) CNTS Workshop July 8 2019 3 / 11

  7. Transportation Queues jobs server signalized intersection freeway (w/ CAVs) mobility on demand jobs: pickup/delivery requests jobs: vehicles jobs: vehicles server: vehicle fleet server: intersection server: freeway infrastructure Ketan Savla (USC) CNTS Workshop July 8 2019 3 / 11

  8. Transportation Queues jobs server signalized intersection freeway (w/ CAVs) mobility on demand jobs: pickup/delivery requests jobs: vehicles jobs: vehicles server: vehicle fleet server: intersection server: freeway infrastructure Service paradigms determined by automation and control Ketan Savla (USC) CNTS Workshop July 8 2019 3 / 11

  9. Performance Evaluation λ capacity ? wait time ? server Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

  10. Performance Evaluation λ capacity ? wait time ? server Constant Service Rate λ − c = queue growth rate ���� service rate Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

  11. Performance Evaluation λ capacity ? wait time ? server Constant Service Rate λ − c = queue growth rate ���� service rate capacity = c Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

  12. Performance Evaluation λ capacity ? wait time ? server Constant Service Rate λ − c = queue growth rate ���� service rate capacity = c Example: M/M/1 Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

  13. Performance Evaluation λ capacity ? wait time ? server Constant Service Rate λ − c = queue growth rate ���� service rate capacity = c Example: M/M/1 c ≡ c ( queue length ) Ketan Savla (USC) CNTS Workshop July 8 2019 4 / 11

  14. � � State Dependent Transportation Queues capacity wait time λ Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

  15. � � State Dependent Transportation Queues capacity wait time λ Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

  16. � � State Dependent Transportation Queues capacity wait time λ Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

  17. � � State Dependent Transportation Queues capacity wait time overestimate λ Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

  18. � � State Dependent Transportation Queues capacity wait time λ Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

  19. � � State Dependent Transportation Queues capacity wait time underestimate λ Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

  20. � � State Dependent Transportation Queues capacity wait time spatial queue vs λ Θ( λ/m 2 ) Θ( λ 2 /m 3 ) Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11

  21. State Dependent Transportation Queues capacity wait time spatial queue vs λ Θ( λ/m 2 ) Θ( λ 2 /m 3 ) vacation queue 10 Vacation Queue Webster Model Vacation Queue (Time Average) 8 Uninterrupted Model Average Queue Length Akcelik Model 6 Link 2 4 Link 1 2 0 0 100 200 300 400 Time Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11 � �

  22. State Dependent Transportation Queues capacity wait time spatial queue vs λ Θ( λ/m 2 ) Θ( λ 2 /m 3 ) vacation queue processor sharing queue 10 Vacation Queue Webster Model Vacation Queue (Time Average) 8 Uninterrupted Model Average Queue Length Akcelik Model 6 Link 2 vs 4 Link 1 2 0 0 100 200 300 400 Time Ketan Savla (USC) CNTS Workshop July 8 2019 5 / 11 � �

  23. Current Notions of Capacity Traffic Capacity [Highway Capacity Manual] “ . . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . . ” Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

  24. Current Notions of Capacity Traffic Capacity [Highway Capacity Manual] “ . . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . . ” “ . . . rate at which . . . vehicles can traverse an intersection approach . . . assuming that the green signal is available at all times . . . ” Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

  25. Current Notions of Capacity Traffic Capacity [Highway Capacity Manual] “ . . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . . ” “ . . . rate at which . . . vehicles can traverse an intersection approach . . . assuming that the green signal is available at all times . . . ” f ≤ c Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

  26. Current Notions of Capacity Traffic Capacity [Highway Capacity Manual] “ . . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . . ” “ . . . rate at which . . . vehicles can traverse an intersection approach . . . assuming that the green signal is available at all times . . . ” f ≤ c c − f : local robustness Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

  27. Current Notions of Capacity Current capacity notions are local Traffic Capacity [Highway Capacity Manual] “ . . . maximum number of vehicles that can pass a given point . . . (assuming) no influence from downstream traffic operation . . . ” “ . . . rate at which . . . vehicles can traverse an intersection approach . . . assuming that the green signal is available at all times . . . ” f ≤ c c − f : local robustness Ketan Savla (USC) CNTS Workshop July 8 2019 6 / 11

  28. Towards Network Capacity network capacity : ( { c i } , physical constraints , control ) Ketan Savla (USC) CNTS Workshop July 8 2019 7 / 11

  29. Dynamical Network Flow λ out ( t ) λ Mass Conservation f i x = λ + R T ( x ) f ( x, u ) ˙ − f ( x, u ) x i � �� � � �� � inflow outflow x i : queue on link i R ( x ) : routing matrix Ketan Savla (USC) CNTS Workshop July 8 2019 8 / 11

  30. Dynamical Network Flow λ out ( t ) λ Mass Conservation f i x = λ + R T ( x ) f ( x, u ) ˙ − f ( x, u ) x i � �� � � �� � inflow outflow x i : queue on link i equilibrium x ∗ : λ out ( t ) = λ R ( x ) : routing matrix existence, stability, and robustness of x ∗ Ketan Savla (USC) CNTS Workshop July 8 2019 8 / 11

  31. Distributed Feedback Control � T min J ( x ( t ) , u ( t )) dt u 0 subj. to x = traffic flow dynamics ˙ u ≡ ramp metering, variable speed limit, routing Ketan Savla (USC) CNTS Workshop July 8 2019 9 / 11

  32. Distributed Feedback Control � T min J ( x ( t ) , u ( t )) dt u 0 subj. to x = traffic flow dynamics ˙ u ≡ ramp metering, variable speed limit, routing open-loop: u ( t ) exact convex relaxation distributed computation Ketan Savla (USC) CNTS Workshop July 8 2019 9 / 11

  33. Distributed Feedback Control � T min J ( x ( t ) , u ( t )) dt u 0 subj. to x = traffic flow dynamics ˙ u ≡ ramp metering, variable speed limit, routing open-loop: u ( t ) exact convex relaxation distributed computation feedback: u ( x ) [ThC02.3] principled distributed control global computation of u ( . ) Ketan Savla (USC) CNTS Workshop July 8 2019 9 / 11

  34. Distributed Feedback Control � T min J ( x ( t ) , u ( t )) dt u 0 subj. to x = traffic flow dynamics ˙ u ≡ ramp metering, variable speed limit, routing open-loop: u ( t ) exact convex relaxation distributed computation feedback: u ( x ) [ThC02.3] principled distributed control global computation of u ( . ) Ketan Savla (USC) CNTS Workshop July 8 2019 9 / 11

  35. 𝑕 2 [𝑙] From State to Output Feedback Control 𝜐 1 𝜐 2 𝜐 3 ̃ 2 [𝑙] = ∑ 3 𝑕 𝜐 𝑗 𝑗=1 direct access to x not available 𝑕 2 [𝑙] 𝑕 2 [𝑙] y : detector measurement 𝜐 2 𝜐 3 𝜐 1 𝜐 2 𝜐 3 𝜐 1 𝜐 4 ̃ 2 [𝑙] = ∑ 𝑕 3 𝜐 𝑗 𝑕 ̃ 2 [𝑙] = ∑ 4 𝜐 𝑗 𝑗=1 𝑗=1 𝑕 2 [𝑙] 𝜐 2 𝜐 3 𝜐 1 𝜐 4 𝑕 ̃ 2 [𝑙] = ∑ 4 𝜐 𝑗 𝑗=1 Ketan Savla (USC) CNTS Workshop July 8 2019 10 / 11

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