✵✳ ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❧❡r ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ❲♦◆❡❈❛ ✷✵✶✻ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✵✴✷✵
✶✳ ■♥tr♦❞✉❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ▼●❋ ❈❛❧❝✉❧✉s ✸ ❋✐rst ❙✉❝❝❡ss❡s ✹ ❙◆❈ ❛t ❲♦r❦ ✺ ❈♦♥❝❧✉s✐♦♥ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✴✷✵
✶✳ ■♥tr♦❞✉❝t✐♦♥ ❲❤❛t ✐s ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❄ Flow of Packets Throttle A B C ∧ U C Feedback-Loop D Σ ∗ Flow of ACKs ▼♦st ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡✿ ✇✐♥❞♦✇✲❜❛s❡❞ tr❛♥s♣♦rt ♣r♦t♦❝♦❧s ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✷✴✷✵
✶✳ ■♥tr♦❞✉❝t✐♦♥ ❊♥❞✲t♦✲❡♥❞ ❉❡s❝r✐♣t✐♦♥ Throttle A B C 𝑉 ∧ U Feedback-Loop C D Σ ∗ n = ✵ U ( n ) U ∧ = ¯ U fb := � ∞ fb ( t ) ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✸✴✷✵
■❢ ✐s s✉❜❛❞❞✐t✐✈❡ ✐ts s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡ ❡q✉❛❧s ✶ ✳ ❲❤❡r❡ ✶ ✐s t❤❡ ♥❡✉tr❛❧ ❡❧❡♠❡♥t ♦❢ t❤❡ ♠✐♥✲♣❧✉s ❝♦♥✈♦❧✉t✐♦♥✳ ✭ ✶ ❢♦r ❛❧❧ ✵✮✳ ❈♦♥✈♦❧✉t✐♦♥ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ s✉❜❛❞❞✐t✐✈✐t②✳ ❚❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❧❡❢t♦✈❡r s❡r✈✐❝❡ ❝✉r✈❡s ❞♦❡s ♥♦t ♣r❡s❡r✈❡ s✉❜❛❞❞✐t✐✈✐t②✳ ❘✉❧❡ ♦❢ t❤✉♠❜✿ ✏■♥t❡r❡st✐♥❣ t❤✐♥❣s ❛r❡ ♥♦t s✉❜❛❞❞✐t✐✈❡✳✑ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❙✉❜❛❞❞✐t✐✈✐t② ❉❡✜♥✐t✐♦♥ U ✐s s✉❜❛❞❞✐t✐✈❡ ✐❢ U ( s ) + U ( t ) ≥ U ( t + s ) � � U ( s , r ) + U ( r , t ) ≥ U ( s , t ) ✏❚❤❡ ✇❤♦❧❡ ✐s s♠❛❧❧❡r t❤❛♥ t❤❡ s✉♠ ♦❢ ✐ts ♣❛rts✑ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✹✴✷✵
❈♦♥✈♦❧✉t✐♦♥ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ s✉❜❛❞❞✐t✐✈✐t②✳ ❚❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❧❡❢t♦✈❡r s❡r✈✐❝❡ ❝✉r✈❡s ❞♦❡s ♥♦t ♣r❡s❡r✈❡ s✉❜❛❞❞✐t✐✈✐t②✳ ❘✉❧❡ ♦❢ t❤✉♠❜✿ ✏■♥t❡r❡st✐♥❣ t❤✐♥❣s ❛r❡ ♥♦t s✉❜❛❞❞✐t✐✈❡✳✑ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❙✉❜❛❞❞✐t✐✈✐t② ❉❡✜♥✐t✐♦♥ U ✐s s✉❜❛❞❞✐t✐✈❡ ✐❢ U ( s ) + U ( t ) ≥ U ( t + s ) � � U ( s , r ) + U ( r , t ) ≥ U ( s , t ) ✏❚❤❡ ✇❤♦❧❡ ✐s s♠❛❧❧❡r t❤❛♥ t❤❡ s✉♠ ♦❢ ✐ts ♣❛rts✑ ■❢ U ✐s s✉❜❛❞❞✐t✐✈❡ ✐ts s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡ ❡q✉❛❧s U ∧ ✶ ✳ ❲❤❡r❡ ✶ ✐s t❤❡ ♥❡✉tr❛❧ ❡❧❡♠❡♥t ♦❢ t❤❡ ♠✐♥✲♣❧✉s ❝♦♥✈♦❧✉t✐♦♥✳ ✭ ✶ ( t ) = ∞ ❢♦r ❛❧❧ t > ✵✮✳ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✹✴✷✵
✶✳ ■♥tr♦❞✉❝t✐♦♥ ❙✉❜❛❞❞✐t✐✈✐t② ❉❡✜♥✐t✐♦♥ U ✐s s✉❜❛❞❞✐t✐✈❡ ✐❢ U ( s ) + U ( t ) ≥ U ( t + s ) � � U ( s , r ) + U ( r , t ) ≥ U ( s , t ) ✏❚❤❡ ✇❤♦❧❡ ✐s s♠❛❧❧❡r t❤❛♥ t❤❡ s✉♠ ♦❢ ✐ts ♣❛rts✑ ■❢ U ✐s s✉❜❛❞❞✐t✐✈❡ ✐ts s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡ ❡q✉❛❧s U ∧ ✶ ✳ ❲❤❡r❡ ✶ ✐s t❤❡ ♥❡✉tr❛❧ ❡❧❡♠❡♥t ♦❢ t❤❡ ♠✐♥✲♣❧✉s ❝♦♥✈♦❧✉t✐♦♥✳ ✭ ✶ ( t ) = ∞ ❢♦r ❛❧❧ t > ✵✮✳ ❈♦♥✈♦❧✉t✐♦♥ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ s✉❜❛❞❞✐t✐✈✐t②✳ ❚❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❧❡❢t♦✈❡r s❡r✈✐❝❡ ❝✉r✈❡s ❞♦❡s ♥♦t ♣r❡s❡r✈❡ s✉❜❛❞❞✐t✐✈✐t②✳ ❘✉❧❡ ♦❢ t❤✉♠❜✿ ✏■♥t❡r❡st✐♥❣ t❤✐♥❣s ❛r❡ ♥♦t s✉❜❛❞❞✐t✐✈❡✳✑ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✹✴✷✵
✷✳ ▼●❋ ❈❛❧❝✉❧✉s ▼●❋ ❈❛❧❝✉❧✉s ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ▼●❋ ❈❛❧❝✉❧✉s ✸ ❋✐rst ❙✉❝❝❡ss❡s ✹ ❙◆❈ ❛t ❲♦r❦ ✺ ❈♦♥❝❧✉s✐♦♥ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✺✴✷✵
■♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s t❤❡ ▼●❋ ✵ ♠✉st ❜❡ ❜♦✉♥❞❡❞✳ ❍♦✇❡✈❡r✿ ✳ ✵ ✷✳ ▼●❋ ❈❛❧❝✉❧✉s ▼●❋✲❇♦✉♥❞s ▼●❋✲❇♦✉♥❞s ❧✐✈❡ ♦♥ ❛ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ♥❡t✇♦r❦ ❝❛❧❝✉❧✉s✳ ❚❤r♦tt❧❡✬s s❡r✈✐❝❡ ✐s st✐❧❧ t❤❡ s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡ ¯ U fb ✳ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✻✴✷✵
❍♦✇❡✈❡r✿ ✳ ✵ ✷✳ ▼●❋ ❈❛❧❝✉❧✉s ▼●❋✲❇♦✉♥❞s ▼●❋✲❇♦✉♥❞s ❧✐✈❡ ♦♥ ❛ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ♥❡t✇♦r❦ ❝❛❧❝✉❧✉s✳ ❚❤r♦tt❧❡✬s s❡r✈✐❝❡ ✐s st✐❧❧ t❤❡ s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡ ¯ U fb ✳ ■♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s t❤❡ ▼●❋ ∞ � U fb ⊗ U )( s , t ) � � e θ A ⊘ ( U ( n ) fb ⊗ U )( s , t ) � e θ A ⊘ ( ¯ � ≤ E E n = ✵ ♠✉st ❜❡ ❜♦✉♥❞❡❞✳ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✻✴✷✵
✷✳ ▼●❋ ❈❛❧❝✉❧✉s ▼●❋✲❇♦✉♥❞s ▼●❋✲❇♦✉♥❞s ❧✐✈❡ ♦♥ ❛ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ♥❡t✇♦r❦ ❝❛❧❝✉❧✉s✳ ❚❤r♦tt❧❡✬s s❡r✈✐❝❡ ✐s st✐❧❧ t❤❡ s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡ ¯ U fb ✳ ■♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s t❤❡ ▼●❋ ∞ � U fb ⊗ U )( s , t ) � � e θ A ⊘ ( U ( n ) fb ⊗ U )( s , t ) � e θ A ⊘ ( ¯ � ≤ E E n = ✵ ♠✉st ❜❡ ❜♦✉♥❞❡❞✳ � fb ⊗ U )( s , t ) � e θ A ⊘ ( U ( n ) ❍♦✇❡✈❡r✿ � ∞ = ∞ ✳ n = ✵ E ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✻✴✷✵
✸✳ ❋✐rst ❙✉❝❝❡ss❡s ❋✐rst ❙✉❝❝❡ss❡s ✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ▼●❋ ❈❛❧❝✉❧✉s ✸ ❋✐rst ❙✉❝❝❡ss❡s ✹ ❙◆❈ ❛t ❲♦r❦ ✺ ❈♦♥❝❧✉s✐♦♥ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✼✴✷✵
∧ Σ ✸✳ ❋✐rst ❙✉❝❝❡ss❡s ▼❡t❤♦❞ ✵✿ ❇✐✈❛r✐❛t❡ ❈❛❧❝✉❧✉s ❙✉❜❛❞❞✐t✐✈✐t② ❧❡❛❞s t♦ ❛ ❝❧♦s❡❞ ❢♦r♠ ♦❢ t❤❡ s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡✿ ¯ U fb = U fb ∧ ✶ ✳ ❋✐rst ✐❞❡❛✿ ❈♦♥s✐❞❡r s✉❜❛❞❞✐t✐✈❡ ❢❡❡❞❜❛❝❦ ❧♦♦♣s✳ ■♥ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥s ❧❡❢t♦✈❡r s❡r✈✐❝❡ ❞❡s❝r✐♣t✐♦♥s ♣r❡s❡r✈❡ s✉❜❛❞❞✐t✐✈✐t② ✭♥♦ ❛rr✐✈❛❧ ❝✉r✈❡s ✐♥✈♦❧✈❡❞✮✳ ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✽✴✷✵
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