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Proportionally fair scheduling for traffic light networks


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Presentation from NORTHMOST - a new biannual series of meetings on the topic of mathematical modelling in transport.

Hosted at, NORTHMOST 01 focussed on academic research, to encourage networking and collaboration between academics interested in the methodological development of mathematical modelling applied to transport.

The focus of the meetings will alternate; NORTHMOST 02 - planned for Spring 2017 - will be led by practitioners who are modelling experts. Practitioners will give presentations, with academic researchers in the audience. In addition to giving a forum for expert practitioners to meet and share best practice, a key aim of the series is to close the gap between research and practice, establishing a feedback loop to communicate the needs of practitioners to those working in university research.

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Proportionally fair scheduling for traffic light networks

  1. 1. PROPORTIONALLY FAIR SCHEDULING FOR TRAFFIC LIGHT NETWORKS Neil Walton University of Manchester Joint work with Peter Kovacs, Tung Le, Rudesindo Núñez-Queija, Hai Vu.
  2. 2. Urban road traffic
  3. 3. Urban road traffic • Densely populated urban areas • Increasing demand • Policy Objectives: Decentralized, optimal, stable, adaptive, scalable, non-anticipative
  4. 4. Outline I. Proportionally fair policy II. Choice of cycle lengths A. The square root rule B. Connection with the capacity region III. Stability results
  5. 5. I. Proportionally fair policy – notation     Road network:
  6. 6. I. Proportionally fair policy – cycles                    
  7. 7. I. Proportionally fair policy – service Setup phase       Linear phase
  8. 8. I. Proportionally fair policy – control • Cycle lengths – in advance • Proportions allocated to phases – cycle to cycle Restrictions: • Every phase needs to be enacted • Every switch requires a switching period of constant length
  9. 9. I. Proportionally fair policy • Estimate the expected queue lengths, • Determine cycle lengths for each junction by the square root rule: • Allocate green times by the optimization problem
  10. 10. II. Choice of cycle length Trade-off between capacity and average waiting times: • Shorter cycles provide shorter average waiting times in a stable system • Longer cycles provide broader capacity region What is the optimal scaling of cycle lengths?
  11. 11. II.A The square root rule Polling model for a single junction:        
  12. 12. II.A The square root rule Use the following notation for the expected cycle length, Introduce condition which imposes similarity to proportional fairness: Stability condition:
  13. 13. II.A The square root rule Formula for the expected queue lengths as a function of the expected cycle length, • PF-condition • Little’s Law: • Relation:
  14. 14. II.A The square root rule – symmetric case  
  15. 15. II.A The square root rule – heavy traffic  
  16. 16. II.B Network capacity Possible schedules     Load in queue 1 Load in queue 2 Problems: • Admissible set of rates < Capacity region? • Convexity?
  17. 17. II.B Network capacity • Switching times and setups decrease the set of admissible rates • In longer cycles these effects are present to a lesser extent • We can find sufficient cycle lengths where these problems vanish:
  18. 18. III. Stability results – routes          
  19. 19. III. Stability results – dynamics • Route-wise accounting for queueing dynamics: • External arrivals are assumed to be Poisson on every route, thus they are Poisson for every in-road with
  20. 20. III. Stability results – fluid limit With the assumption that vehicles on separate routes are distributed homogeneously on the in-roads the fluid limit is as follows:
  21. 21. III. Stability results – main theorem Proof: by Lyapunov-function.