Presentation from NORTHMOST - a new biannual series of meetings on the topic of mathematical modelling in transport. Hosted at its.leeds.ac.uk, 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.

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. Urban road traffic

3. Urban road traffic
• Densely populated urban areas
• Increasing demand
• Policy Objectives: Decentralized,
optimal, stable, adaptive,
scalable, non-anticipative

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. I. Proportionally fair policy –
notation
Road network:

6. I. Proportionally fair policy –
cycles

7. I. Proportionally fair policy –
service
Setup phase
Linear phase

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. 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. 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. II.A The square root rule
Polling model for a single junction:

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. 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. II.A The square root rule –
symmetric case

15. II.A The square root rule –
heavy traffic

16. II.B Network capacity
Possible
schedules
Load in
queue 1
Load in
queue 2
Problems:
• Admissible set of rates < Capacity
region?
• Convexity?

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. III. Stability results – routes

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. 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. III. Stability results – main
theorem
Proof: by Lyapunov-function.

22. THANK YOU
FOR YOUR
ATTENTION!