Flötteröd, Gunnar

Infoscience - École polytechnique fédérale de Lausanne

Neessary and suient ondition for unique owsin ertain intersetion modelsGunnar Flötteröd1May 12, 20111in the ontext of joint work with R. Corthout, C. Tampere, F. Viti1 / 23Motivation•deterministi dynami tra assignment•ambiguous solutions at network level possible•what about ambiguous solutions at node level?2 / 23OutlineModeling assumptionsSuient onditionNeessary onditionSummary and outlook3 / 23OutlineModeling assumptionsSuient onditionNeessary onditionSummary and outlook4 / 23Modeling assumptions (1/2)∆1∆2...∆IΣ1Σ2...ΣJ• ∆iis ow demand of ingoing link i• Σjis ow supply of outgoing link j•not every i needs to ompete for every j•also overs internal onits5 / 23Modeling assumptions (2/2)•every ingoing link i maximizes its inow qinisubjet to demand onstraints qini≤ ∆i supply onstraints qoutj≤ Σjon node outows qoutj rst-in/rst-out and ow onservation:qoutj=∑iβijqiniwith turning frations βij qi1and qi2are onstrained by j ⇒ qi1/qi2= αi1j/αi2jwith stritly positive and nite priorities αij•priorities are deisive for unique ow solutions6 / 23OutlineModeling assumptionsSuient onditionNeessary onditionSummary and outlook7 / 23Suient ondition and solution algorithm•Flows are unique if there are positive αiand jsuh thatαij= αijholds for all upstream links i that ould enter onstraint j .•Proof: By known solution algorithm.1. assign unique inow priority αi= αij/jto every ingoing link2. set all inows to zero; label all i = 1 . . . I as unonstrained3. while there are unonstrained inows left:3.1 inrease all unonstrained inows proportionally to theirpriorities until the next onstraint binds3.2 label all inows that reahed the onstraint as onstrained8 / 23OutlineModeling assumptionsSuient onditionNeessary onditionSummary and outlook9 / 23Neessary ondition•The suient ondition is also neessary:There are positive αiand jsuh thatαij= αijholds for all upstream links i that ould enter onstraint j .•Sketh of proof: Assume that the neessary ondition does not hold. Show that a non-unique solution an always be onstruted.10 / 23Preliminaries•Every outgoing link j with only one ingoing link i an betransformed into a demand onstraint on i .•Hene, for all aeted i and j :1. replae ∆iby min{∆i,Σj/βij}2. remove j from onsideration11 / 23Test priority•A test priority α˜(j1, . . . , jJ) is dened as follows:α˜(j1, . . . , jI) =α1j1...αIjI j1, . . . jIarbitrary•That is, go through all ingoing links i = 1 . . . I and assign to itone of its priorities αij.12 / 23Linearly dependent test prios ⇔ neessary ond.•The neessary ondition is equivalent to linear dependene ofall test priorities:αij= αij∀i , j ⇔ α˜(j1, . . . , jI) ∝ α˜(l1, . . . , lI) ∀ji, li•Proof: ⇒ by insertion; ⇐ by rearrangement.•Impliation: If the neessary ondition does not hold, there are linearly independent test priorities α˜Aand α˜B; there are ingoing links i16= i2with α˜Ai1/α˜Ai26= α˜Bi1/α˜Bi2.•Fous on these links.13 / 23Setting of boundary onditions•Consider two supply onstraints:Σj1≥ βi1,j1qi1 + βi2,j1qi2 +∑l 6=i1,i2βl ,j1qlΣj2≥ βi1,j2qi1 + βi2,j2qi2 +∑l 6=i1,i2βj ,j2ql.•Sale down Σj1, Σj2, and all ∆iuntil all other Σjnever bind.•Ensure that i1, i2are the rst inows to be onstrained.14 / 23Three ases1. both i1and i2ompete for both j1and j22. i1does not ompete for j23. i1does not ompete for j1and i2does not ompete for j215 / 23Case 1: both i ompete for both jqi1qi2αAi2αAi1αBi2αBi1onstr. j2or j1onstr. j1or j216 / 23Constrution of ase 1•binding onstraints:qi2=1βi2,j1Σj1−∑l 6=i1,i2βl ,j1ql− βi1,j1βi2,j1qi1qi2=1βi2,j2Σj2−∑l 6=i1,i2βj ,j2ql− βi1,j2βi2,j2qi1•onstrution:1. set onstraint slopes with the βs2. shift onstraints with the Σs17 / 23Case 2: i1does not ompete for j2qi1qi2αAi2αAi1αBi2αBi1onstr. j2at solution Bonstr. j1at solution A18 / 23Constrution of ase 2•bindung onstraints:qi2=1βi2,j1Σj1−∑l 6=i1,i2βl ,j1ql− βi1,j1βi2,j1qi1qi2=1βi2,j2Σj2−∑l 6=i1,i2βj ,j2ql•onstrution:1. set the β for some slope of j12. set the Σ suh that j2is atop of j119 / 23Case 3: i1(i2) ompetes only for j1(j2)qi1qi2αAi2αAi1αBi2αBi1onstr. j2at solution Bonstr. j1at solution A20 / 23Constrution of ase 3•binding onstraints:qi1=1βi1,j1Σj1−∑l 6=i1,i2βl ,j1qlqi2=1βi2,j2Σj2−∑l 6=i1,i2βj ,j2ql•onstrution:1. set the Σ somehow2. avoid unique solution by reduing some ∆l, l 6= i1, i2until itshifts a binding onstraint21 / 23OutlineModeling assumptionsSuient onditionNeessary onditionSummary and outlook22 / 23Summary and outlook•summary: better keep the node model simple•outlook: nonlinear internal node onstraints23 / 23

Necessary and sufficient condition for unique flows in certain intersection models

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Necessary and sufficient condition for unique flows in certain intersection models

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