proportionally fair scheduling for traffic light networks
TRANSCRIPT
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PROPORTIONALLY FAIR SCHEDULING
FOR TRAFFIC LIGHT NETWORKS
Neil WaltonUniversity of Manchester
Joint work with Peter Kovacs, Tung Le, Rudesindo Núñez-Queija, Hai Vu.
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Urban road traffic
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Urban road traffic
• Densely populated urban areas
• Increasing demand
• Policy Objectives: Decentralized,
optimal, stable, adaptive,
scalable, non-anticipative
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OutlineI. Proportionally fair policy
II. Choice of cycle lengths
A. The square root rule
B. Connection with the capacity
region
III. Stability results
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I. Proportionally fair policy – notation
Road network:
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I. Proportionally fair policy – cycles
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I. Proportionally fair policy – service
Setup phase
Linear phase
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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
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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
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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?
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II.A The square root rule
Polling model for a single junction:
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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:
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II.A The square root ruleFormula for the expected queue lengths as a function of the expected cycle length,
• PF-condition
• Little’s Law:
• Relation:
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II.A The square root rule – symmetric case
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II.A The square root rule – heavy traffic
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II.B Network capacity
Possible schedules
Load in queue 1
Load in queue 2
Problems:• Admissible set of rates < Capacity
region?• Convexity?
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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:
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III. Stability results – routes
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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
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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:
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III. Stability results – main theorem
Proof: by Lyapunov-function.
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THANK YOU FOR YOUR
ATTENTION!