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✵✳

❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥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

∧

U ∗ Σ A B C D

Feedback-Loop

C

Throttle

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✐♦♥

𝑉∧ U ∗ Σ A B C D

Feedback-Loop

C

Throttle

U∧ = ¯ Ufb := ∞

n=✵ U(n) fb (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✑

■❢ ✐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✐✈❡✳✑

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐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 ❙✳ ✹✴✷✵

✶✳ ■♥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 ❙✳ ✺✴✷✵

✷✳ ▼●❋ ❈❛❧❝✉❧✉s

▼●❋✲❇♦✉♥❞s

▼●❋✲❇♦✉♥❞s ❧✐✈❡ ♦♥ ❛ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ♥❡t✇♦r❦ ❝❛❧❝✉❧✉s✳ ❚❤r♦tt❧❡✬s s❡r✈✐❝❡ ✐s st✐❧❧ t❤❡ s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡ ¯ Ufb✳ ■♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s t❤❡ ▼●❋

✵

♠✉st ❜❡ ❜♦✉♥❞❡❞✳ ❍♦✇❡✈❡r✿

✵

✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐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❡ ¯ Ufb✳ ■♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s t❤❡ ▼●❋ E

• eθA⊘( ¯

Ufb⊗U)(s,t)

≤

∞

• n=✵

E

• eθA⊘(U(n)

fb ⊗U)(s,t)

♠✉st ❜❡ ❜♦✉♥❞❡❞✳ ❍♦✇❡✈❡r✿

✵

✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐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❡ ¯ Ufb✳ ■♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s t❤❡ ▼●❋ E

• eθA⊘( ¯

Ufb⊗U)(s,t)

≤

∞

• n=✵

E

• eθA⊘(U(n)

fb ⊗U)(s,t)

♠✉st ❜❡ ❜♦✉♥❞❡❞✳ ❍♦✇❡✈❡r✿ ∞

n=✵ E

• eθA⊘(U(n)

fb ⊗U)(s,t)

= ∞✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐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 ❙✳ ✼✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✵✿ ❇✐✈❛r✐❛t❡ ❈❛❧❝✉❧✉s

❙✉❜❛❞❞✐t✐✈✐t② ❧❡❛❞s t♦ ❛ ❝❧♦s❡❞ ❢♦r♠ ♦❢ t❤❡ s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡✿ ¯ Ufb = Ufb ∧ ✶✳ ❋✐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 ✐♥✈♦❧✈❡❞✮✳

∧

Σ

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✽✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✵✿ ❇✐✈❛r✐❛t❡ ❈❛❧❝✉❧✉s

❙✉❜❛❞❞✐t✐✈✐t② ❧❡❛❞s t♦ ❛ ❝❧♦s❡❞ ❢♦r♠ ♦❢ t❤❡ s✉❜❛❞❞✐t✐✈❡ ❝❧♦s✉r❡✿ ¯ Ufb = Ufb ∧ ✶✳ ❋✐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 ✐♥✈♦❧✈❡❞✮✳

∧

Uk 1 Σ A B C D

Feedback-Loop

C

Throttle

Bx

Uk(s, t) = k(t − s)

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✽✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✶✿ ❈❤❛♥❣❡ ❚♦♣♦❧♦❣✐❡s

P❛rt ♦❢ ✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s✑ ✭❚❘✱ ✷✵✶✺✮ ❇② ❝❤❛♥❣✐♥❣ t❤❡ t♦♣♦❧♦❣② s✉❜❛❞❞✐t✐✈✐t② ❝❛♥ ❜❡ ❡♥❢♦r❝❡❞✳ ❈❛♥ ❜❡ ❝♦st❧②✳

∧

Σ

∧

Σ

◆❡✈❡rt❤❡❧❡ss✿ ✜rst ♥♦♥✲tr✐✈✐❛❧ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s ♦♥ ❲❋❈✲❙②st❡♠s✦ ❲♦r❦s ❜❡tt❡r ✐❢✿

s❡r✈✐❝❡ ❡❧❡♠❡♥ts ❤❛✈❡ s✐♠✐❧❛r r❛t❡s ❡❛❝❤ s❡r✈✐❝❡ ❡❧❡♠❡♥t ❝❛♥ ✇♦r❦ t❤❡ ❡♥t✐r❡ ❛❣❣r❡❣❛t❡ ♦❢ ❝r♦ss✢♦✇s

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✾✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✶✿ ❈❤❛♥❣❡ ❚♦♣♦❧♦❣✐❡s

P❛rt ♦❢ ✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s✑ ✭❚❘✱ ✷✵✶✺✮ ❇② ❝❤❛♥❣✐♥❣ t❤❡ t♦♣♦❧♦❣② s✉❜❛❞❞✐t✐✈✐t② ❝❛♥ ❜❡ ❡♥❢♦r❝❡❞✳ ❈❛♥ ❜❡ ❝♦st❧②✳

∧

U V Σ A B C D

Feedback-Loop

C

Throttle

Bx Cx

∧

U V Σ A B C D

Feedback-Loop

C

Throttle

Bx+Cx

◆❡✈❡rt❤❡❧❡ss✿ ✜rst ♥♦♥✲tr✐✈✐❛❧ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s ♦♥ ❲❋❈✲❙②st❡♠s✦ ❲♦r❦s ❜❡tt❡r ✐❢✿

s❡r✈✐❝❡ ❡❧❡♠❡♥ts ❤❛✈❡ s✐♠✐❧❛r r❛t❡s ❡❛❝❤ s❡r✈✐❝❡ ❡❧❡♠❡♥t ❝❛♥ ✇♦r❦ t❤❡ ❡♥t✐r❡ ❛❣❣r❡❣❛t❡ ♦❢ ❝r♦ss✢♦✇s

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✾✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✶✿ ❈❤❛♥❣❡ ❚♦♣♦❧♦❣✐❡s

P❛rt ♦❢ ✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s✑ ✭❚❘✱ ✷✵✶✺✮ ❇② ❝❤❛♥❣✐♥❣ t❤❡ t♦♣♦❧♦❣② s✉❜❛❞❞✐t✐✈✐t② ❝❛♥ ❜❡ ❡♥❢♦r❝❡❞✳ ❈❛♥ ❜❡ ❝♦st❧②✳

∧

U V Σ A B C D

Feedback-Loop

C

Throttle

Bx Cx

∧

U V Σ A B C D

Feedback-Loop

C

Throttle

Bx+Cx

◆❡✈❡rt❤❡❧❡ss✿ ✜rst ♥♦♥✲tr✐✈✐❛❧ ♣❡r❢♦r♠❛♥❝❡ ❜♦✉♥❞s ♦♥ ❲❋❈✲❙②st❡♠s✦ ❲♦r❦s ❜❡tt❡r ✐❢✿

s❡r✈✐❝❡ ❡❧❡♠❡♥ts ❤❛✈❡ s✐♠✐❧❛r r❛t❡s ❡❛❝❤ s❡r✈✐❝❡ ❡❧❡♠❡♥t ❝❛♥ ✇♦r❦ t❤❡ ❡♥t✐r❡ ❛❣❣r❡❣❛t❡ ♦❢ ❝r♦ss✢♦✇s ▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✾✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✷✿ ✏❆ss✉♠❡✑ ❙✉❜❛❞❞✐t✐✈✐t②

✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❚❤❡

• ❡♥❡r❛❧ ❈❛s❡✑ ✭❱❛❧✉❡t♦♦❧s ✷✵✶✺✮

❆ ❝♦♥❞✐t✐♦♥ ❢♦r s✉❜❛❞❞✐t✐✈✐t②✿ (E) : U ⊗ V (s, t) − (U ⊗ V )(✷)(s, t) ≤ Σ ❢♦r ❛❧❧ s ≤ t ⇒ Ufb ✐s s✉❜❛❞❞✐t✐✈❡✳ ■❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ t❤❡ ❲❋❈ s②st❡♠ ❝❛♥ ❜❡ ❛♥❛❧②③❡❞ ❛s ✇❡❧❧ ✐s s✉❜❛❞❞✐t✐✈❡ ❲♦r❦s ❜❡tt❡r ❢♦r

❞✐✛❡r✐♥❣ s❡r✈✐❝❡ ❡❧❡♠❡♥ts

❚❤❡ ❝❛t❝❤✿ ❞✐✈❡r❣❡s ✐♥ ✦

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✵✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✷✿ ✏❆ss✉♠❡✑ ❙✉❜❛❞❞✐t✐✈✐t②

✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❚❤❡

• ❡♥❡r❛❧ ❈❛s❡✑ ✭❱❛❧✉❡t♦♦❧s ✷✵✶✺✮

❆ ❝♦♥❞✐t✐♦♥ ❢♦r s✉❜❛❞❞✐t✐✈✐t②✿ (E) : U ⊗ V (s, t) − (U ⊗ V )(✷)(s, t) ≤ Σ ❢♦r ❛❧❧ s ≤ t ⇒ Ufb ✐s s✉❜❛❞❞✐t✐✈❡✳ ■❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ (¬E) ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ t❤❡ ❲❋❈ s②st❡♠ ❝❛♥ ❜❡ ❛♥❛❧②③❡❞ ❛s ✇❡❧❧ ✐s s✉❜❛❞❞✐t✐✈❡ ❲♦r❦s ❜❡tt❡r ❢♦r

❞✐✛❡r✐♥❣ s❡r✈✐❝❡ ❡❧❡♠❡♥ts

❚❤❡ ❝❛t❝❤✿ ❞✐✈❡r❣❡s ✐♥ ✦

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✵✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✷✿ ✏❆ss✉♠❡✑ ❙✉❜❛❞❞✐t✐✈✐t②

✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❚❤❡

• ❡♥❡r❛❧ ❈❛s❡✑ ✭❱❛❧✉❡t♦♦❧s ✷✵✶✺✮

❆ ❝♦♥❞✐t✐♦♥ ❢♦r s✉❜❛❞❞✐t✐✈✐t②✿ (E) : U ⊗ V (s, t) − (U ⊗ V )(✷)(s, t) ≤ Σ ❢♦r ❛❧❧ s ≤ t ⇒ Ufb ✐s s✉❜❛❞❞✐t✐✈❡✳ ■❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ (¬E) ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ t❤❡ ❲❋❈ s②st❡♠ ❝❛♥ ❜❡ ❛♥❛❧②③❡❞ ❛s ✇❡❧❧ P(dsys(t) > T) ≤ P(dsys(t) > T | Ufb ✐s s✉❜❛❞❞✐t✐✈❡) + P(¬E) ❲♦r❦s ❜❡tt❡r ❢♦r

❞✐✛❡r✐♥❣ s❡r✈✐❝❡ ❡❧❡♠❡♥ts

❚❤❡ ❝❛t❝❤✿ ❞✐✈❡r❣❡s ✐♥ ✦

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✵✴✷✵

✸✳ ❋✐rst ❙✉❝❝❡ss❡s

▼❡t❤♦❞ ✷✿ ✏❆ss✉♠❡✑ ❙✉❜❛❞❞✐t✐✈✐t②

✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❚❤❡

• ❡♥❡r❛❧ ❈❛s❡✑ ✭❱❛❧✉❡t♦♦❧s ✷✵✶✺✮

❆ ❝♦♥❞✐t✐♦♥ ❢♦r s✉❜❛❞❞✐t✐✈✐t②✿ (E) : U ⊗ V (s, t) − (U ⊗ V )(✷)(s, t) ≤ Σ ❢♦r ❛❧❧ s ≤ t ⇒ Ufb ✐s s✉❜❛❞❞✐t✐✈❡✳ ■❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ (¬E) ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ t❤❡ ❲❋❈ s②st❡♠ ❝❛♥ ❜❡ ❛♥❛❧②③❡❞ ❛s ✇❡❧❧ P(dsys(t) > T) ≤ P(dsys(t) > T | Ufb ✐s s✉❜❛❞❞✐t✐✈❡) + P(¬E) ❲♦r❦s ❜❡tt❡r ❢♦r

❞✐✛❡r✐♥❣ s❡r✈✐❝❡ ❡❧❡♠❡♥ts

❚❤❡ ❝❛t❝❤✿ P(¬E) ❞✐✈❡r❣❡s ✐♥ t✦

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✵✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

❙◆❈ ❛t ❲♦r❦

✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ▼●❋ ❈❛❧❝✉❧✉s ✸ ❋✐rst ❙✉❝❝❡ss❡s ✹ ❙◆❈ ❛t ❲♦r❦ ✺ ❈♦♥❝❧✉s✐♦♥

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✶✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

• ❡♥❡r❛❧ ❈❛s❡

❋r♦♠ ✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❚❤❡

• ❡♥❡r❛❧ ❈❛s❡✑ ✭❱❛❧✉❡t♦♦❧s ✷✵✶✺✮

❚❤r♦tt❧❡❞ ✈s✳ ❯♥t❤r♦tt❧❡❞✿

∧

U V Σ A B C D

Feedback-Loop

C

Throttle

Bx Cx

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✷✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

• ❡♥❡r❛❧ ❈❛s❡ ✭t❤r♦tt❧❡❞ ✈s✳ ✉♥t❤r♦tt❧❡❞✮

❋r♦♠ ✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❚❤❡

• ❡♥❡r❛❧ ❈❛s❡✑ ✭❱❛❧✉❡t♦♦❧s ✷✵✶✺✮

■✳✐✳❞✳ ❡①♣(λ) ✐♥❝r❡♠❡♥ts ❢♦r ❛❧❧ ✢♦✇s ✐♥✈♦❧✈❡❞✳

0.3 0.4 0.5 0.6 0.7 0.8 1e−16 1e−12 1e−08 1e−04 1e+00 Utilization in % Violation Prob. Sigma = 10 Sigma = 15 Sigma = 20

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✸✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

❈♦♥✈❡r❣❡♥❝❡ t♦ ❯♥t❤r♦tt❧❡❞ ❙②st❡♠s❄

❋r♦♠ ✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❚❤❡

• ❡♥❡r❛❧ ❈❛s❡✑ ✭❱❛❧✉❡t♦♦❧s ✷✵✶✺✮

❇♦✉♥❞s ❢♦r ✐♠♣r♦✈✐♥❣ ✇✐♥❞♦✇ s✐③❡s Σ✿

5 10 15 20 25 30 35 40 1e−06 1e−04 1e−02 1e+00 Window Sigma Violation Prob.

v = 1.1 v = 1.5 v = 2

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✹✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

❉◆❈ ✈s✳ ❚♦♣♦❧♦❣② ❈❤❛♥❣❡s ✈s✳ P(¬E)

❋r♦♠ ✏❙t♦❝❤❛st✐❝ ❲♦rst ❈❛s❡ ❆♥❛❧②s✐s ♦❢ ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❧❡❞ ❙②st❡♠s✑ ✭✉♥❞❡r r❡✈✐❡✇✮

• ✐✈❡♥ t❤❡ s❛♠❡ s❝❡♥❛r✐♦ ✇❡ ❝❛♥ ❛♥❛❧②③❡ ✐t ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ♦r ❜② t❤❡

t✇♦ st♦❝❤❛st✐❝ ♠❡t❤♦❞s✳ ❆❞♠✐ss✐♦♥ ♣r♦❜❧❡♠✿

❆❞♠✐tt✐♥❣ ♦♥❡ s✉❜✢♦✇ ♦❢ ✐♥❝r❡❛s❡s ✉t✐❧✐③❛t✐♦♥ ❛t ❜② ✶✪✳ ✐s ✉t✐❧✐③❡❞ ❜② ❛ ❝❡rt❛✐♥ ❛♠♦✉♥t ♦❢ ❝r♦ss✲✢♦✇s ❛❧r❡❛❞②✳ ❍♦✇ ♠❛♥② ✢♦✇s ❝❛♥ ✇❡ ❛❞♠✐t t♦ t❤❡ s②st❡♠ ✇✐t❤♦✉t ❜r❡❛❦✐♥❣ ❛ ❣✐✈❡♥ ✭♣r♦❜❛❜✐❧✐st✐❝✮ ❜❛❝❦❧♦❣ ❜♦✉♥❞ ❛t t❤❡ t❤r♦tt❧❡ ❡❧❡♠❡♥t❄

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✺✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

❉◆❈ ✈s✳ ❚♦♣♦❧♦❣② ❈❤❛♥❣❡s ✈s✳ P(¬E)

❋r♦♠ ✏❙t♦❝❤❛st✐❝ ❲♦rst ❈❛s❡ ❆♥❛❧②s✐s ♦❢ ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❧❡❞ ❙②st❡♠s✑ ✭✉♥❞❡r r❡✈✐❡✇✮

• ✐✈❡♥ t❤❡ s❛♠❡ s❝❡♥❛r✐♦ ✇❡ ❝❛♥ ❛♥❛❧②③❡ ✐t ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ♦r ❜② t❤❡

t✇♦ st♦❝❤❛st✐❝ ♠❡t❤♦❞s✳ ❆❞♠✐ss✐♦♥ ♣r♦❜❧❡♠✿

❆❞♠✐tt✐♥❣ ♦♥❡ s✉❜✢♦✇ ♦❢ A ✐♥❝r❡❛s❡s ✉t✐❧✐③❛t✐♦♥ ❛t U ❜② ✶✪✳ U ✐s ✉t✐❧✐③❡❞ ❜② ❛ ❝❡rt❛✐♥ ❛♠♦✉♥t ♦❢ ❝r♦ss✲✢♦✇s ❛❧r❡❛❞②✳

❍♦✇ ♠❛♥② ✢♦✇s ❝❛♥ ✇❡ ❛❞♠✐t t♦ t❤❡ s②st❡♠ ✇✐t❤♦✉t ❜r❡❛❦✐♥❣ ❛ ❣✐✈❡♥ ✭♣r♦❜❛❜✐❧✐st✐❝✮ ❜❛❝❦❧♦❣ ❜♦✉♥❞ ❛t t❤❡ t❤r♦tt❧❡ ❡❧❡♠❡♥t❄

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✺✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

❉◆❈ ✈s✳ ❚♦♣♦❧♦❣② ❈❤❛♥❣❡s ✈s✳ P(¬E)

❋r♦♠ ✏❙t♦❝❤❛st✐❝ ❲♦rst ❈❛s❡ ❆♥❛❧②s✐s ♦❢ ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❧❡❞ ❙②st❡♠s✑ ✭✉♥❞❡r r❡✈✐❡✇✮

• ✐✈❡♥ t❤❡ s❛♠❡ s❝❡♥❛r✐♦ ✇❡ ❝❛♥ ❛♥❛❧②③❡ ✐t ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ♦r ❜② t❤❡

t✇♦ st♦❝❤❛st✐❝ ♠❡t❤♦❞s✳ ❆❞♠✐ss✐♦♥ ♣r♦❜❧❡♠✿

❆❞♠✐tt✐♥❣ ♦♥❡ s✉❜✢♦✇ ♦❢ A ✐♥❝r❡❛s❡s ✉t✐❧✐③❛t✐♦♥ ❛t U ❜② ✶✪✳ U ✐s ✉t✐❧✐③❡❞ ❜② ❛ ❝❡rt❛✐♥ ❛♠♦✉♥t ♦❢ ❝r♦ss✲✢♦✇s ❛❧r❡❛❞②✳ ❍♦✇ ♠❛♥② ✢♦✇s ❝❛♥ ✇❡ ❛❞♠✐t t♦ t❤❡ s②st❡♠ ✇✐t❤♦✉t ❜r❡❛❦✐♥❣ ❛

❣✐✈❡♥ ✭♣r♦❜❛❜✐❧✐st✐❝✮ ❜❛❝❦❧♦❣ ❜♦✉♥❞ ❛t t❤❡ t❤r♦tt❧❡ ❡❧❡♠❡♥t❄

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✺✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

❉◆❈ ✈s✳ ❚♦♣♦❧♦❣② ❈❤❛♥❣❡s ✈s✳ P(¬E)

❋r♦♠ ✏❙t♦❝❤❛st✐❝ ❲♦rst ❈❛s❡ ❆♥❛❧②s✐s ♦❢ ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❧❡❞ ❙②st❡♠s✑ ✭✉♥❞❡r r❡✈✐❡✇✮ ❘❡❛❝❤❛❜❧❡ ✉t✐❧✐③❛t✐♦♥s✿ ❉◆❈ ✭✉♣ t♦ ✺✷✪✮✱ ❜② t♦♣♦❧♦❣② ❝❤❛♥❣❡ ✭✉♣ t♦ ✻✺✪✮✱ ❛♥❞ ❜② ❜♦✉♥❞✐♥❣ P(¬E) ✭✉♣ t♦ ✾✵✪✱ t = ✶✵✵✵✮

20 40 60 80 Utilization Flows N 0.01 0.25 0.50 0.75 0.01 0.25 0.50 0.75 0.01 0.25 0.50 0.75 10-3 10-6 10-9 10-3 10-6 10-9

❆s②♠♠❡tr✐❝ s✐t✉❛t✐♦♥✿ V r✉♥s ❢❛st❡r t❤❛♥ U✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✻✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

❉◆❈ ✈s✳ ❚♦♣♦❧♦❣② ❈❤❛♥❣❡s ✈s✳ P(¬E)

❋r♦♠ ✏❙t♦❝❤❛st✐❝ ❲♦rst ❈❛s❡ ❆♥❛❧②s✐s ♦❢ ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❧❡❞ ❙②st❡♠s✑ ✭✉♥❞❡r r❡✈✐❡✇✮ ◆♦✇ V ❛♥❞ U ❤❛✈❡ t❤❡ s❛♠❡ r❛t❡✳ V ❤❛♥❞❧❡s ✶✵ ❝r♦ss✢♦✇s✳ ❋♦r ❛ ❧♦✇ ♥✉♠❜❡r ♦❢ ❝r♦ss✢♦✇s ❛t t❤❡ s❡r✈✐❝❡ ❡❧❡♠❡♥ts ❛r❡ ✈❡r② s✐♠✐❧❛r✦ ❜♦✉♥❞✐♥❣ ❜❡❝♦♠❡s ❤❛r❞❡r✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✼✴✷✵

✹✳ ❙◆❈ ❛t ❲♦r❦

❉◆❈ ✈s✳ ❚♦♣♦❧♦❣② ❈❤❛♥❣❡s ✈s✳ P(¬E)

❋r♦♠ ✏❙t♦❝❤❛st✐❝ ❲♦rst ❈❛s❡ ❆♥❛❧②s✐s ♦❢ ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❧❡❞ ❙②st❡♠s✑ ✭✉♥❞❡r r❡✈✐❡✇✮ ◆♦✇ V ❛♥❞ U ❤❛✈❡ t❤❡ s❛♠❡ r❛t❡✳ V ❤❛♥❞❧❡s ✶✵ ❝r♦ss✢♦✇s✳

20 40 60 80 Utilization Flows N 0.01 0.25 0.50 0.75 0.01 0.25 0.50 0.75 0.01 0.25 0.50 0.75 0.01 0.25 0.50 0.75 10-3 10-9 t = 10 t = 100

❋♦r ❛ ❧♦✇ ♥✉♠❜❡r ♦❢ ❝r♦ss✢♦✇s ❛t U t❤❡ s❡r✈✐❝❡ ❡❧❡♠❡♥ts ❛r❡ ✈❡r② s✐♠✐❧❛r✦ →❜♦✉♥❞✐♥❣ P(¬E) ❜❡❝♦♠❡s ❤❛r❞❡r✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✼✴✷✵

✺✳ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

✶ ■♥tr♦❞✉❝t✐♦♥ ✷ ▼●❋ ❈❛❧❝✉❧✉s ✸ ❋✐rst ❙✉❝❝❡ss❡s ✹ ❙◆❈ ❛t ❲♦r❦ ✺ ❈♦♥❝❧✉s✐♦♥

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✽✴✷✵

✺✳ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❲❋❈ ✐s ❛ ❤❛r❞ ♣r♦❜❧❡♠ ✐♥ ❙◆❈✳ ❘❡❝❡♥t ✇♦r❦ ✐♥ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥s ❣✐✈❡s ✜rst r❡s✉❧ts✱ t❤♦✉❣❤✳ ❈✉rr❡♥t ❛♣♣r♦❛❝❤❡s ❡✐t❤❡r✿

❝♦♥s✐❞❡r s♣❡❝✐✜❝ ❢❡❡❞❜❛❝❦ ❧♦♦♣s ♦♥❧② ✭✜①❡❞ ❞❡❧❛② ❡❧❡♠❡♥ts✮ ✉♥❞❡r❡st✐♠❛t❡ ❛✈❛✐❧❛❜❧❡ s❡r✈✐❝❡ r❛t❡s ✭t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s✮ ❞❡♣❡♥❞ ♦♥ t❤❡ ❡✈❛❧✉❛t✐♦♥ t✐♠❡ ✭❜♦✉♥❞✐♥❣ ✮

▼❡t❤♦❞s ❝♦♠♣❧❡♠❡♥t ♦♥❡ ❛♥♦t❤❡r t♦ s♦♠❡ ❡①t❡♥t✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✾✴✷✵

✺✳ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❲❋❈ ✐s ❛ ❤❛r❞ ♣r♦❜❧❡♠ ✐♥ ❙◆❈✳ ❘❡❝❡♥t ✇♦r❦ ✐♥ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥s ❣✐✈❡s ✜rst r❡s✉❧ts✱ t❤♦✉❣❤✳ ❈✉rr❡♥t ❛♣♣r♦❛❝❤❡s ❡✐t❤❡r✿

❝♦♥s✐❞❡r s♣❡❝✐✜❝ ❢❡❡❞❜❛❝❦ ❧♦♦♣s ♦♥❧② ✭✜①❡❞ ❞❡❧❛② ❡❧❡♠❡♥ts✮

✉♥❞❡r❡st✐♠❛t❡ ❛✈❛✐❧❛❜❧❡ s❡r✈✐❝❡ r❛t❡s ✭t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s✮ ❞❡♣❡♥❞ ♦♥ t❤❡ ❡✈❛❧✉❛t✐♦♥ t✐♠❡ ✭❜♦✉♥❞✐♥❣ ✮

▼❡t❤♦❞s ❝♦♠♣❧❡♠❡♥t ♦♥❡ ❛♥♦t❤❡r t♦ s♦♠❡ ❡①t❡♥t✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✾✴✷✵

✺✳ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❲❋❈ ✐s ❛ ❤❛r❞ ♣r♦❜❧❡♠ ✐♥ ❙◆❈✳ ❘❡❝❡♥t ✇♦r❦ ✐♥ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥s ❣✐✈❡s ✜rst r❡s✉❧ts✱ t❤♦✉❣❤✳ ❈✉rr❡♥t ❛♣♣r♦❛❝❤❡s ❡✐t❤❡r✿

❝♦♥s✐❞❡r s♣❡❝✐✜❝ ❢❡❡❞❜❛❝❦ ❧♦♦♣s ♦♥❧② ✭✜①❡❞ ❞❡❧❛② ❡❧❡♠❡♥ts✮ ✉♥❞❡r❡st✐♠❛t❡ ❛✈❛✐❧❛❜❧❡ s❡r✈✐❝❡ r❛t❡s ✭t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s✮

❞❡♣❡♥❞ ♦♥ t❤❡ ❡✈❛❧✉❛t✐♦♥ t✐♠❡ ✭❜♦✉♥❞✐♥❣ ✮

▼❡t❤♦❞s ❝♦♠♣❧❡♠❡♥t ♦♥❡ ❛♥♦t❤❡r t♦ s♦♠❡ ❡①t❡♥t✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✾✴✷✵

✺✳ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❲❋❈ ✐s ❛ ❤❛r❞ ♣r♦❜❧❡♠ ✐♥ ❙◆❈✳ ❘❡❝❡♥t ✇♦r❦ ✐♥ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥s ❣✐✈❡s ✜rst r❡s✉❧ts✱ t❤♦✉❣❤✳ ❈✉rr❡♥t ❛♣♣r♦❛❝❤❡s ❡✐t❤❡r✿

❝♦♥s✐❞❡r s♣❡❝✐✜❝ ❢❡❡❞❜❛❝❦ ❧♦♦♣s ♦♥❧② ✭✜①❡❞ ❞❡❧❛② ❡❧❡♠❡♥ts✮ ✉♥❞❡r❡st✐♠❛t❡ ❛✈❛✐❧❛❜❧❡ s❡r✈✐❝❡ r❛t❡s ✭t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s✮ ❞❡♣❡♥❞ ♦♥ t❤❡ ❡✈❛❧✉❛t✐♦♥ t✐♠❡ t ✭❜♦✉♥❞✐♥❣ P(¬E)✮

▼❡t❤♦❞s ❝♦♠♣❧❡♠❡♥t ♦♥❡ ❛♥♦t❤❡r t♦ s♦♠❡ ❡①t❡♥t✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✾✴✷✵

✺✳ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❲❋❈ ✐s ❛ ❤❛r❞ ♣r♦❜❧❡♠ ✐♥ ❙◆❈✳ ❘❡❝❡♥t ✇♦r❦ ✐♥ ❜✐✈❛r✐❛t❡ ❢♦r♠✉❧❛t✐♦♥s ❣✐✈❡s ✜rst r❡s✉❧ts✱ t❤♦✉❣❤✳ ❈✉rr❡♥t ❛♣♣r♦❛❝❤❡s ❡✐t❤❡r✿

❝♦♥s✐❞❡r s♣❡❝✐✜❝ ❢❡❡❞❜❛❝❦ ❧♦♦♣s ♦♥❧② ✭✜①❡❞ ❞❡❧❛② ❡❧❡♠❡♥ts✮ ✉♥❞❡r❡st✐♠❛t❡ ❛✈❛✐❧❛❜❧❡ s❡r✈✐❝❡ r❛t❡s ✭t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s✮ ❞❡♣❡♥❞ ♦♥ t❤❡ ❡✈❛❧✉❛t✐♦♥ t✐♠❡ t ✭❜♦✉♥❞✐♥❣ P(¬E)✮

▼❡t❤♦❞s ❝♦♠♣❧❡♠❡♥t ♦♥❡ ❛♥♦t❤❡r t♦ s♦♠❡ ❡①t❡♥t✳

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✶✾✴✷✵

✺✳ ❈♦♥❝❧✉s✐♦♥

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❋♦r ❞❡t❛✐❧s✿ ✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s✑ ✭❇❡❝❦ ❛♥❞ ❙❝❤♠✐tt✳ ❚❘✱ ✷✵✶✺✮ ✏❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧ ✐♥ ❙t♦❝❤❛st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❚❤❡

• ❡♥❡r❛❧ ❈❛s❡✑ ✭❇❡❝❦ ❛♥❞ ❙❝❤♠✐tt✱ ❱❛❧✉❡t♦♦❧s ✷✵✶✺✮

✏❙t♦❝❤❛st✐❝ ❲♦rst ❈❛s❡ ❆♥❛❧②s✐s ♦❢ ❲✐♥❞♦✇ ❋❧♦✇ ❈♦♥tr♦❧❧❡❞ ❙②st❡♠s✑ ✭❇❡❝❦✳ ❯♥❞❡r r❡✈✐❡✇✮

▼✐❝❤❛❡❧ ❆✳ ❇❡❝❦ ❚❯ ❑❛✐s❡rs❧❛✉t❡r♥ ✕ ❲❋❈ ❛♥❞ ❙✉❜❛❞❞✐t✐✈✐t② ❙✳ ✷✵✴✷✵