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❉❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❞✉❛❧ ❢♦r♠✉❧❛t✐♦♥ ✐♥ tr❛✣❝ ❛ss✐❣♥♠❡♥t ♣r♦❜❧❡♠s ▲✐s❛♥❞r♦ ❆✳ P❛r❡♥t❡ ❈■❋❆❙■❙ ✲ ❈❖◆■❈❊❚ ✲❯♥✐✈❡rs✐❞❛❞ ◆❛❝✐♦♥❛❧ ❞❡ ❘r♦s❛r✐♦ ✲ ❆r❣❡♥t✐♥❛✱ ♣❛r❡♥t❡❅❝✐❢❛s✐s✲❝♦♥✐❝❡t✳❣♦✈✳❛r ❥♦✐♥t ✇♦r❦ ✇✐t❤ ▲✳ ❈♦rr❛❧❡s ❛♥❞ P✳ ❆✳ ▲♦t✐t♦ ❱■ ▲❛t✐♥ ❆♠❡r✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❈♦♥tr♦❧ ◗✉✐t♦ ✲ ❊❝✉❛❞♦r ❙❡♣t❡♠❜❡r ✸✲✼✱ ✷✵✶✽

■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ▼♦t✐✈❛t✐♦♥ ❚r❛✣❝ ❛ss✐❣♥♠❡♥t ♣r♦❜❧❡♠s✿ ♣r✐♠❛❧ ❢♦r♠✉❧❛t✐♦♥s ■♥✈❡rs❡ ♣r♦❜❧❡♠✿ ❞❡♠❛♥❞ ❛❞❥✉st♠❡♥t ♣r♦❜❧❡♠ ❉✉❛❧ ❛♣♣r♦❛❝❤ ✷ ❉✉❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r tr❛✣❝ ❡q✉✐❧✐❜r✐✉♠ ❙♣❧✐tt✐♥❣ ❛♣♣r♦❛❝❤ ❉✉❛❧ ❞❡♠❛♥❞ ❛❞❥✉st♠❡♥t ♣r♦❜❧❡♠

■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❈♦♥❣❡st✐♦♥ ◆✉♠❜❡r ♦❢ ✈❡❤✐❝❧❡s ✐♥❝r❡❛s❡s ❜✉t ❛✈❛✐❧❛❜❧❡ r♦✉t❡ s♣❛❝❡ ✐s ❛❧♠♦st ❝♦♥st❛♥t

❆❧❧ t❤❡s❡ ♣r♦❜❧❡♠s ♥❡❡❞ ❛ ❝♦♠♠♦♥ ✐♥♣✉t✿ ❚r❛✣❝ ❉❡♠❛♥❞ ❛♥❞ ❛ ♠♦❞❡❧ ♦❢ ❞r✐✈❡r ❜❡❤❛✈✐♦r ■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❚r❛✣❝ Pr♦❜❧❡♠s ❚r❛♥s♣♦rt ◆❡t✇♦r❦ ❞❡s✐❣♥ ■♠♣❛❝t ♦❢ ✐♥❢r❛str✉❝t✉r❡ ♠♦❞✐✜❝❛t✐♦♥s ♦♥ tr❛✣❝ ❞✐str✐❜✉t✐♦♥ ✭s❤♦rt t❡r♠✮✱ ■♠♣❛❝t ♦❢ t❤❡s❡ ♠♦❞✐✜❝❛t✐♦♥s ♦♥ tr❛✣❝ ❞❡♠❛♥❞ ✭❧♦♥❣ t❡r♠✮✱ ■♠♣❛❝t ♦❢ t❤♦s❡ ♠♦❞✐✜❝❛t✐♦♥s ♦♥ ❤♦✉s❡❤♦❧❞ ❞✐str✐❜✉t✐♦♥s ✭❡✈❡♥ ❧♦♥❣❡r t❡r♠✮✳ ❚r❛✣❝ r❡❣✉❧❛t✐♦♥ ✭❝♦♥tr♦❧✮ ❘❡❛❧ t✐♠❡ tr❛✣❝ st❛t❡ ❡st✐♠❛t✐♦♥ ❚r❛✣❝ ❧✐❣❤t s❡tt✐♥❣ ✜①❡❞ ♦r ❞②♥❛♠✐❝ ❘❡❛❧ t✐♠❡ r❡❛❝t✐♦♥ ♦❢ ✈❡❤✐❝❧❡s ✇✳r✳t✳ ❝❤❛♥❣❡s ✐♥ s②st❡♠ ❝♦♥❞✐t✐♦♥s✳ ❚r❛✣❝ ❋♦r❡❝❛st✐♥❣ ●✉✐❞❛♥❝❡

■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❚r❛✣❝ Pr♦❜❧❡♠s ❚r❛♥s♣♦rt ◆❡t✇♦r❦ ❞❡s✐❣♥ ■♠♣❛❝t ♦❢ ✐♥❢r❛str✉❝t✉r❡ ♠♦❞✐✜❝❛t✐♦♥s ♦♥ tr❛✣❝ ❞✐str✐❜✉t✐♦♥ ✭s❤♦rt t❡r♠✮✱ ■♠♣❛❝t ♦❢ t❤❡s❡ ♠♦❞✐✜❝❛t✐♦♥s ♦♥ tr❛✣❝ ❞❡♠❛♥❞ ✭❧♦♥❣ t❡r♠✮✱ ■♠♣❛❝t ♦❢ t❤♦s❡ ♠♦❞✐✜❝❛t✐♦♥s ♦♥ ❤♦✉s❡❤♦❧❞ ❞✐str✐❜✉t✐♦♥s ✭❡✈❡♥ ❧♦♥❣❡r t❡r♠✮✳ ❚r❛✣❝ r❡❣✉❧❛t✐♦♥ ✭❝♦♥tr♦❧✮ ❘❡❛❧ t✐♠❡ tr❛✣❝ st❛t❡ ❡st✐♠❛t✐♦♥ ❚r❛✣❝ ❧✐❣❤t s❡tt✐♥❣ ✜①❡❞ ♦r ❞②♥❛♠✐❝ ❘❡❛❧ t✐♠❡ r❡❛❝t✐♦♥ ♦❢ ✈❡❤✐❝❧❡s ✇✳r✳t✳ ❝❤❛♥❣❡s ✐♥ s②st❡♠ ❝♦♥❞✐t✐♦♥s✳ ❚r❛✣❝ ❋♦r❡❝❛st✐♥❣ ●✉✐❞❛♥❝❡ ❆❧❧ t❤❡s❡ ♣r♦❜❧❡♠s ♥❡❡❞ ❛ ❝♦♠♠♦♥ ✐♥♣✉t✿ ❚r❛✣❝ ❉❡♠❛♥❞ ❛♥❞ ❛ ♠♦❞❡❧ ♦❢ ❞r✐✈❡r ❜❡❤❛✈✐♦r

■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❚r❛✣❝ ♠♦❞❡❧ G = ( N , A ) ❞✐r❡❝t❡❞ ❣r❛♣❤ ✭tr❛✣❝ ♥❡t✇♦r❦✮ D ⊂ N s❡t ♦❢ ❞❡st✐♥❛t✐♦♥s g d i ≥ ✵ ❞❡♠❛♥❞ ❢r♦♠ i ∈ N t♦ d ∈ D ✇✐t❤ i � = d R d i s❡t ♦❢ r♦✉t❡s ✭s✐♠♣❧❡ ♣❛t❤s✮ ❢r♦♠ i ∈ N t♦ d ∈ D ✇✐t❤ i � = d i � = d R d R = � i x r ≥ ✵ ✢♦✇ ♦♥ r♦✉t❡ r ∈ R ✱ s❛t✐s❢②✐♥❣ g d i = � i x r . r ∈ R d f a t♦t❛❧ ✢♦✇ ♦♥ ❛r❝ a ∈ A ✱ s❛t✐s❢②✐♥❣ f a = � r ∋ a x r ✳ t a = s a ( f a ) tr❛✈❡❧ t✐♠❡ ♦♥ ❛r❝ a ❋✐❣✉r❡✿ ❙✐♦✉① ❋❛❧❧s tr❛✣❝ ♥❡t✇♦r❦ ✇✐t❤ s♦♠❡ ❞❡♠❛♥❞s

■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❉r✐✈❡r ❇❡❤❛✈✐♦r ❯s❡r ❊q✉✐❧✐❜r✐✉♠ ❆♥② ❞r✐✈❡r ✇✐❧❧ ❝❤❛♥❣❡ t♦ ❛ s❤♦rt❡r ✭❢❛st❡r✮ ♣❛t❤✳ ❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐s ❛❝❤✐❡✈❡❞ ✇❤❡♥ ❛❧❧ ✉s❡❞ ♣❛t❤ ❤❛s t❤❡ s❛♠❡ tr❛✈❡❧ t✐♠❡ ❛ ♥♦ ♣❛t❤ ❤❛s ❛ s❤♦rt❡r tr❛✈❡❧ t✐♠❡✳ ❲❛r❞r♦♣ ❊q✉✐❧✐❜r✐✉♠ ❋♦r ❛♥② i ∈ N , d ∈ D ✱ ❛❧❧ t❤❡ ✉s❡❞ r♦✉t❡s ♦❢ R d i ❤❛✈❡ t❤❡ s❛♠❡ tr❛✈❡❧ t✐♠❡ ❛♥❞ ♥♦ r♦✉t❡ ❤❛s ✭str✐❝t❧②✮ ❧♦✇❡r tr❛✈❡❧ t✐♠❡✱ ✐✳❡✳ x r ∈ R d � t a = τ d ✇❤❡r❡ τ d � i , x r > ✵ = ⇒ i , i = ♠✐♥ t a . r ∈ R d a ∈ r i a ∈ r t ❚r❛✈❡❧ t✐♠❡ ❢✉♥❝t✐♦♥s ❛r❡ str✐❝t❧② free-flow ✐♥❝r❡❛s✐♥❣ ❢r♦♠ R t♦ R + ✳ queuing ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❇P❘ ❢✉♥❝t✐♦♥ ✐s 2 t 0 jamming t 0 s ( f ) = t ✵ ( ✶ + ( f / c ) α ) c 0 f

■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❚r❛✣❝ ❛ss✐❣♥♠❡♥t ♣r♦❜❧❡♠✿ r♦✉t❡✲✢♦✇ ❢♦r♠✉❧❛t✐♦♥ ❙♦♠❡ ❛ss✉♠♣t✐♦♥s ❚❤❡ ♥❡t✇♦r❦ ✐s str♦♥❣❧② ❝♦♥♥❡❝t❡❞✳ ❚❤❡ tr❛✈❡❧ t✐♠❡ ♦❢ ❛ r♦✉t❡ ✐s t❤❡ s✉♠ ♦❢ t❤❡ tr❛✈❡❧ t✐♠❡s ♦✈❡r t❤❡ ❛r❝s✳ ❚❤❡ tr❛✈❡❧ t✐♠❡ ♦❢ ❛♥ ❛r❝ ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ ✢♦✇ tr♦✉❣❤ t❤✐s ❛r❝ ❛♥❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ✢♦✇ tr♦✉❣❤ ❛♥② ♦t❤❡r ❛r❝✳ ❚❤❡ tr❛✈❡❧ t✐♠❡ ❢✉♥❝t✐♦♥s ❛r❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣✳ ❇❡❝❦♠❛♥✬s ❢♦r♠✉❧❛t✐♦♥ ❢♦ ❲❛r❞r♦♣ ❡q✳ � f a � ♠✐♥ s a ( z ) dz ✵ a ∈ A s . t . ✭✶✮ g d � i = x r , x r ≥ ✵ , r ∈ R d i � f a = x r . r ∋ a

■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❆r❝✲✢♦✇ ❢♦r♠✉❧❛t✐♦♥ ❉✐ss❛❣r❡❣❛t❡❞ ❛r❝✲✢♦✇ ✈❛r✐❛❜❧❡s y d a ≥ ✵ ✢♦✇ ♦♥ ❛r❝ a ✇✐t❤ ❞❡st✐♥❛t✐♦♥ d ❘❡❧❛①❡❞ ✭❜✉t ❡q✉✐✈❛❧❡♥t✦✦✮ ♣r♦❜❧❡♠ � f a � ♠✐♥ s a ( z ) dz ✵ a ∈ A s . t . ✭✷✮ � � g d y d y d i + a = a , a ∈ A + a ∈ A − i i � y d y d f a = a , a ≥ ✵ . d ∈ D ❍❡r❡ A − ✐s t❤❡ s❡t ♦❢ ❛r❝s ✇✐t❤ t❡r♠✐♥❛❧ ♥♦❞❡ i ❛♥❞ A + i ✐s t❤❡ s❡t ♦❢ ❛r❝s ✇✐t❤ i ✐♥✐t✐❛❧ ♥♦❞❡ i ✳

■♥tr♦❞✉❝t✐♦♥ ❉✉❛❧ ❛♣♣r♦❛❝❤ ❇✐❧❡✈❡❧ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❞❡♠❛♥❞ ❡st✐♠❛t✐♦♥ ❖♥ s♦♠❡ ❛r❝s t❤❡ ✢♦✇ ✐s ♦❜s❡r✈❡❞ 4 ✭r❡❞ ❛r❝s✮ 9 Pr♦❜❧❡♠✿ ❋✐♥❞ t❤❡ tr❛✣❝ ❞❡♠❛♥❞ d 14 ✇❤♦s❡ ❛ss✐♥❣♠❡♥t ♠❛t❝❤❡s t❤❡ 1 5 6 7 2 ♦❜s❡r✈❡❞ ✢♦✇✳ ●✐✈❡♥ ˜ A t❤❡ s❡t ♦❢ ♦❜s❡r✈❡❞ ❛r❝s ❧❡t P 8 ❜❡ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ✇❤♦❧❡ ❛r❝ ✢♦✇ ✈❡❝t♦r w = ( f a ) a ∈ A ♦✈❡r t❤❡ d 24 3 ♦❜s❡r✈❡❞ ❛r❝s✳ ■❢ ˜ f ✐s t❤❡ ✈❡❝t♦r ♦❢ ♦❜s❡r✈❡❞ ✢♦✇s ❛♥❞ ˜ g ✐s ❛♥ ♦✉t❞❛t❡❞ ❞❡♠❛♥❞✱ t❤❡ ♣r♦❜❧❡♠ ✐s f � ✷ + β � g − ˜ ♠✐♥ g , f � Pf − ˜ g � ✷ s . t . f = ❲❛r❞r♦♣ ( g )