PROPORTIONALLY FAIR SCHEDULING

FOR TRAFFIC LIGHT NETWORKS

Neil WaltonUniversity of Manchester

Joint work with Peter Kovacs, Tung Le, Rudesindo Núñez-Queija, Hai Vu.

Urban road traffic

Urban road traffic

• Densely populated urban areas

• Increasing demand

• Policy Objectives: Decentralized,

optimal, stable, adaptive,

scalable, non-anticipative

OutlineI. Proportionally fair policy

II. Choice of cycle lengths

A. The square root rule

B. Connection with the capacity

region

III. Stability results

I. Proportionally fair policy – notation

 

 

Road network:

I. Proportionally fair policy – cycles 

       

    

     

I. Proportionally fair policy – service

Setup phase

 

 

 

Linear phase

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

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

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?

II.A The square root rule

Polling model for a single junction:

 

 

 

 

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:

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:

II.A The square root rule – symmetric case

 

II.A The square root rule – heavy traffic

 

II.B Network capacity

Possible schedules

 

  Load in queue 1

Load in queue 2

Problems:• Admissible set of rates < Capacity

region?• Convexity?

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:

III. Stability results – routes 

   

 

 

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

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:

III. Stability results – main theorem

Proof: by Lyapunov-function.

THANK YOU FOR YOUR

ATTENTION!