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1❖◆❊❘❆✱ 2❚❯ ❇r❛✉♥s❝❤✇❡✐❣

❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❝✉♠✉❧❛t✐✈❡ ❝✉r✈❡s ❛♥❞ ❡✈❡♥t str❡❛♠s✿ ❢r♦♠ ♥❡t✇♦r❦ ❝❛❧❝✉❧✉s t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

▼❛r❝ ❇♦②❡r✶✱✯✱ P✐❡rr❡ ❘♦✉①✶✱ ▲❡♦♥✐❡ ❑ö❤❧❡r ✷✱ ❇♦r✐s❧❛✈ ◆✐❦♦❧✐❝ ✷✱ ❘♦❧❢ ❊r♥st✷

✺t❤ ❲♦r❦s❤♦♣ ♦♥ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✭❲♦◆❡❈❛✲✺✮ ❖❝t♦❜❡r ✾t❤ ✷✵✷✵

◆❈✴❈P❆

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s

❚❤❡ ♠♦❞❡❧ ❚❤❡ t♦♦❧s

✸ ❘❡s✉❧ts

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦ P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts ❈P❆ ✐♥t❡❣r❛t✐♦♥ ❆❣❣r❡❣❛t✐♦♥

✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

✶✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ✸ ❘❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

◆❈✴❈P❆ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s

❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s

◆❈ ◆❡t✇♦r❦ ❝❛❧❝✉❧✉s ✉♣♣❡r✴❧♦✇❡r ❛rr✐✈❛❧ ❝✉r✈❡s ✭str✐❝t✮ ♠✐♥✐♠❛❧ ✭♠❛①✐♠❛❧✮ s❡r✈✐❝❡ ❝✉r✈❡ s❤❛♣✐♥❣ ❝✉r✈❡s ❘❚❈ ❘❡❛❧✲❚✐♠❡ ❝❛❧❝✉❧✉s ✉♣♣❡r✴❧♦✇❡r ❛rr✐✈❛❧ ❝✉r✈❡s ✉♣♣❡r✴❧♦✇❡r s❡r✈✐❝❡ ❝✉r✈❡s ❣r❡❡❞② s❤❛♣❡rs ❈P❆ ❈♦♠♣♦s✐t✐♦♥❛❧ P❡r❢♦r♠❛♥❝❡ ❆♥❛❧②s✐s ❡✈❡♥t str❡❛♠ ❡✈❡♥t ❞✐st❛♥❝❡ ❜✉s② ✇✐♥❞♦✇ ❚❤r❡❡ ♠♦❞❡❧s ❘❡❧❛t✐♦♥ ❘❚❈ ↔ ◆❈ ❡q✉✐✈❛❧❡♥❝❡ ❬✶✱ ✷❪ ✉♣ t♦ t❡❝❤♥✐❝❛❧ ❞❡t❛✐❧s ❘❡❧❛t✐♦♥s ❈P❆ ↔ ◆❈ q✉✐t❡ t❤❡ s❛♠❡ ♠♦❞❡❧s ♦❢ ✇♦r❦❧♦❛❞ ❬✸✱ ✹❪ ❞✐✛❡r❡♥t ❛♥❛❧②s✐s ♠❡t❤♦❞s

✷✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s

❚❤❡ ♠♦❞❡❧ ❚❤❡ t♦♦❧s

✸ ❘❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s

❚❤❡ ♠♦❞❡❧ ❚❤❡ t♦♦❧s

✸ ❘❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ ♠♦❞❡❧

❚✇♦ ✢♦✇✴❝♦♠♣♦♥❡♥t ♠♦❞❡❧s

❊✈❡♥t ❙tr❡❛♠✴❈P❆ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s C E′ E C A′ A ❋❧♦✇ ♠♦❞❡❧ E(t)✿ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ✉♣ t♦ t✐♠❡ t A(t)✿ ❛♠♦✉♥t ♦❢ ❞❛t❛ ✉♣ t♦ t✐♠❡ t ❈♦♥tr❛❝t η+, η−✿ ❡✈❡♥t ❛rr✐✈❛❧ ❢✉♥❝✲ t✐♦♥s αu, αl✿ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❛r✲ r✐✈❛❧ ❝✉r✈❡s ∀t, d ≥ 0

E(t + d) − E(t) ≤ η+(d) αl(d) ≤ A(t + d) − A(t) ≤ αu(d) E(t + d) − E(t) ≥ η−(d)

❋❧♦✇ tr❛♥s❢♦r✲ ♠❛t✐♦♥ ❇✉s② ✇✐♥❞♦✇ ❘❡s✐❞✉❛❧ s❡r✈✐❝❡

✸✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ ♠♦❞❡❧

❚❤❡ ❣❧♦❜❛❧ ♣✐❝t✉r❡

✹✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ ♠♦❞❡❧

❆ ♣❛❝❦❡t ❢✉♥❝t✐♦♥ ❛s ❣❛t❡✇❛②

❆rr✐✈❛❧ ❝✉r✈❡ P❛❝❦❡t ❝♦✉♥t ❊✈❡♥t ❝♦✉♥t A : R+ → R+ P : R+ → N E : R+ → N A(t)✿ ❛♠♦✉♥t ♦❢ ❞❛t❛ ✉♣ t♦ t P(d)✿ ♥✉♠❜❡r ♦❢ ❢✉❧❧ ♣❛❝❦❡ts ✐♥ t❤❡ d ✜rst ✏❜✐ts✑ E(t)✿ ♥✉♠❜❡r ♦❢ ❢✉❧❧ ♣❛❝❦❡ts ✉♣ t♦ t P(

NC

• A )

❬✺❪

=

CP A

• E

✺✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ ♠♦❞❡❧

■❧❧✉str❛t✐♦♥

t A(t)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

t E(t)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

❞ P(d)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

❙❝❡♥❛r✐♦✿ ❋✐rst ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶ ❙❡❝♦♥❞ ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶✴✷ ❚❤✐r❞ ♣❛❝❦❡t✿ s✐③❡ ✷✱ t❤r♦✉❣❤♣✉t ✷ ❋♦✉rt❤ ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶

✻✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ ♠♦❞❡❧

■❧❧✉str❛t✐♦♥

t A(t)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

t E(t)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

❞ P(d)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

❙❝❡♥❛r✐♦✿ ❋✐rst ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶ ❙❡❝♦♥❞ ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶✴✷ ❚❤✐r❞ ♣❛❝❦❡t✿ s✐③❡ ✷✱ t❤r♦✉❣❤♣✉t ✷ ❋♦✉rt❤ ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶

✻✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ ♠♦❞❡❧

■❧❧✉str❛t✐♦♥

t A(t)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

t E(t)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

❞ P(d)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

❙❝❡♥❛r✐♦✿ ❋✐rst ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶ ❙❡❝♦♥❞ ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶✴✷ ❚❤✐r❞ ♣❛❝❦❡t✿ s✐③❡ ✷✱ t❤r♦✉❣❤♣✉t ✷ ❋♦✉rt❤ ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶

✻✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ ♠♦❞❡❧

■❧❧✉str❛t✐♦♥

t A(t)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

t E(t)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

❞ P(d)

✶ ✶ ✷ ✷ ✸ ✸ ✹ ✹

❙❝❡♥❛r✐♦✿ ❋✐rst ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶ ❙❡❝♦♥❞ ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶✴✷ ❚❤✐r❞ ♣❛❝❦❡t✿ s✐③❡ ✷✱ t❤r♦✉❣❤♣✉t ✷ ❋♦✉rt❤ ♣❛❝❦❡t✿ s✐③❡ ✶✱ t❤r♦✉❣❤♣✉t ✶

✻✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s

❚❤❡ ♠♦❞❡❧ ❚❤❡ t♦♦❧s

✸ ❘❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ t♦♦❧s

■♥t❡r✈❛❧ ❇♦✉♥❞✐♥❣ P❛✐r ✭■❇P✮

• ❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ ❛rr✐✈❛❧ ❝✉r✈❡s✴❡♥✈❡❧♦♣♣❡✴❡✈❡♥t str❡❛♠s

■♥t❡r✈❛❧ ❇♦✉♥❞✐♥❣ P❛✐r✿ r❡♥❛♠✐♥❣ ♦❢ ❛rr✐✈❛❧ ❝✉r✈❡s✴❡✈❡♥t str❡❛♠ φ = (φ, φ) ✐s ❛♥ ■♥t❡r✈❛❧ ❇♦✉❞✐♥❣ P❛✐r ✭■❇P✮ ♦❢ f ✐✛ ∀t, d ≥ 0 : φ(d) ≤ f(t + d) − f(t) ≤ φ(d) ❙❛♠❡ ♣r♦♣❡rt✐❡s t❤❛♥ ❛rr✐✈❛❧ ❝✉r✈❡s✿ ♠✐♥✐♠✉♠ ✭r❡s♣✳ ♠❛①✐♠✉♠✮ ♦❢ ✉♣♣❡r ✭r❡s♣✳ ❧♦✇❡r✮ ❛rr✐✈❛❧ ❝✉r✈❡s✱ s✉❜✴s✉♣♣❡r✲❛❞❞✐t✐✈❡ ❝❧♦s✉r❡✱ ❡t❝✳

✼✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ❚❤❡ t♦♦❧s

Ps❡✉❞♦✲✐♥✈❡rs❡

x f(x) y f −1(y) y f −1(y) ■♥ ❬✻❪✱ ✷✺ ♣r♦♣❡rt✐❡s ♦♥ ♣s❡✉❞♦✲✐♥✈❡rs❡s✱ ❧✐❦❡ f(x) < y = ⇒ x ≤ f −1(y), ✭✶✮ (f ◦ g) −1 ≤ g −1 ◦ f −1, ✭✷✮ φ

−1(δ) ≤ f −1(y + δ) − f −1(y) ≤ φ −1(δ).

✭✸✮

✽✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ✸ ❘❡s✉❧ts

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦ P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts ❈P❆ ✐♥t❡❣r❛t✐♦♥ ❆❣❣r❡❣❛t✐♦♥

✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ✸ ❘❡s✉❧ts

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦ P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts ❈P❆ ✐♥t❡❣r❛t✐♦♥ ❆❣❣r❡❣❛t✐♦♥

✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

◆❈✴❈P❆ ❘❡s✉❧ts ❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❚❤❡ ❡①♣❡❝t❡❞ r❡s✉❧ts

❢r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦✱ ❛♥❞ ❢♦r ❝♦♠♣❧❡t❡♥❡ss✳ A P E

✾✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❚❤❡ ❡①♣❡❝t❡❞ r❡s✉❧ts

❢r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦✱ ❛♥❞ ❢♦r ❝♦♠♣❧❡t❡♥❡ss✳ A P E (α, α) (π, π)

✾✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❚❤❡ ❡①♣❡❝t❡❞ r❡s✉❧ts

❢r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦✱ ❛♥❞ ❢♦r ❝♦♠♣❧❡t❡♥❡ss✳ A P E (α, α) (π, π) (π ◦ α, π ◦ α)

✾✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❚❤❡ ❡①♣❡❝t❡❞ r❡s✉❧ts

❢r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦✱ ❛♥❞ ❢♦r ❝♦♠♣❧❡t❡♥❡ss✳ A P E (α, α) (π, π) (π ◦ α, π ◦ α) (π, π) (η, η)

✾✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❚❤❡ ❡①♣❡❝t❡❞ r❡s✉❧ts

❢r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦✱ ❛♥❞ ❢♦r ❝♦♠♣❧❡t❡♥❡ss✳ A P E (α, α) (π, π) (π ◦ α, π ◦ α) (π −1 ◦ η, π −1 ◦ η) (π, π) (η, η)

✾✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❚❤❡ ❡①♣❡❝t❡❞ r❡s✉❧ts

❢r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦✱ ❛♥❞ ❢♦r ❝♦♠♣❧❡t❡♥❡ss✳ A P E (α, α) (π, π) (π ◦ α, π ◦ α) (π −1 ◦ η, π −1 ◦ η) (π, π) (η, η) (α, α) (η, η)

✾✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦

❚❤❡ ❡①♣❡❝t❡❞ r❡s✉❧ts

❢r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦✱ ❛♥❞ ❢♦r ❝♦♠♣❧❡t❡♥❡ss✳ A P E (α, α) (π, π) (π ◦ α, π ◦ α) (π −1 ◦ η, π −1 ◦ η) (π, π) (η, η) (α, α) (ηl ◦ α −1, ηr ◦ α −1) (η, η)

✾✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ✸ ❘❡s✉❧ts

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦ P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts ❈P❆ ✐♥t❡❣r❛t✐♦♥ ❆❣❣r❡❣❛t✐♦♥

✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

◆❈✴❈P❆ ❘❡s✉❧ts P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts

P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts

S A, E, P A′, E′, P ′ P❛❝❦❡t✐③❡r✿ st♦r❡ ❜✐ts✱ ✉♣ t♦ ❡♥❞✲♦❢✲♣❛❝❦❡t ✐♥st❛♥t❛♥❡♦✉s ♣❛❝❦❡t ♦✉t♣✉t ♠♦❞❡❧✿ E, P ✉♥❝❤❛♥❣❡❞ A′ := P −1 ◦ P ◦ A E′ := E P ′ := P t A(t) t E(t) ❞ P(d) t A′(t)

✶✵✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts

P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts

S A, E, P A′, E′, P ′ P❛❝❦❡t✐③❡r✿ st♦r❡ ❜✐ts✱ ✉♣ t♦ ❡♥❞✲♦❢✲♣❛❝❦❡t ✐♥st❛♥t❛♥❡♦✉s ♣❛❝❦❡t ♦✉t♣✉t ♠♦❞❡❧✿ E, P ✉♥❝❤❛♥❣❡❞ A′ := P −1 ◦ P ◦ A E′ := E P ′ := P α′ := π −1 ◦ η α′ := π −1 ◦ η

✶✵✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ✸ ❘❡s✉❧ts

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦ P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts ❈P❆ ✐♥t❡❣r❛t✐♦♥ ❆❣❣r❡❣❛t✐♦♥

✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

◆❈✴❈P❆ ❘❡s✉❧ts ❈P❆ ✐♥t❡❣r❛t✐♦♥

❈P❆ ✐♥t❡❣r❛t✐♦♥

❊✈❡♥t str❡❛♠✿ η, η ❇♦✉♥❞✐♥❣ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ✐♥ ❛ t✐♠❡ ✐♥t❡r✈❛❧✳ ❊✈❡♥t ❞✐st❛♥❝❡✿ δ, δ ❇♦✉♥❞✐♥❣ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❡✈❡♥ts✳ ❈♦♥tr✐❜✉t✐♦♥s r❡❧❛t❡❞ t♦ ❝✉r✈❡s✿

❞❡✜♥✐t✐♦♥ ♦❢ ❡✈❡♥t ♦❝❝✉r❡♥❝❡ ❢✉♥❝t✐♦♥ T ❞❡✜♥✐t✐♦♥ ♦❢ δ, δ ❛s ■❇P ♦❢ T r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ δ ↔ η ❛♥❞ δ ↔ η✳

❈♦♥tr✐❜✉t✐♦♥s r❡❧❛t❡❞ t♦ ❛♥❛❧②s✐s✿

r❡✇r✐t✐♥❣ ♦❢ ❜✉s②✲✇✐♥❞♦✇ ❛♥❛❧②s✐s ✇✐t❤ ✏❛rr✐✈❛❧ ❝✉r✈❡✑ ♥♦t❛t✐♦♥s ❛❞❛♣t❛t✐♦♥ t♦ ✈❛r✐❛❜❧❡ ♣❛❝❦❡ts✴✇♦r❦❧♦❛❞

✶✶✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❈P❆ ✐♥t❡❣r❛t✐♦♥

buzy−window fix−point min/max buzy−window fix−point delay+jitter CPA component buzy−window fix−point workload delay+jitter CPA component delay+jitter CPA + η η′ α η η′ η′ α′ η α η ◦ π−1

✶✷✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ✸ ❘❡s✉❧ts

❋r♦♠ ◆❈ t♦ ❈P❆✱ ❛♥❞ ❜❛❝❦ P❛❝❦❡t✐③❡r✿ ❣❡♥❡r❛❧✐s✐♥❣ ♣r❡✈✐♦✉s r❡s✉❧ts ❈P❆ ✐♥t❡❣r❛t✐♦♥ ❆❣❣r❡❣❛t✐♦♥

✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

◆❈✴❈P❆ ❘❡s✉❧ts ❆❣❣r❡❣❛t✐♦♥

❆❣❣r❡❣❛t✐♦♥

S A1, E1, P1 A2, E2, P2 A, E, P ❆❣❣r❡❣❛t✐♦♥✿ ♠✐① ♦❢ ✢♦✇s ✏s✉♠✑ ♦❢ ✢♦✇s ✐s ❛ ✢♦✇ ♥♦ ❞❡❧❛② A := A1 + A2 E := E1 + E2 P(A1 + A2) := P(A1) + P(A2) α := α1 + α2 α := α1 + α2 η := η1 + η2 η := η1 + η2 π :=

• π1 ∗ π2
• π := ⌈π1 ∗ π2⌉

✶✸✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❆❣❣r❡❣❛t✐♦♥

❈❛s❡ st✉❞②

❚✇♦ ❞❛t❛ ✢♦✇s✱ F1, F2✱ ❢r♦♠ S t♦ C ❯s✐♥❣ ❛ ❧✐♥❦ ♦❢ t❤r♦✉❣❤♣✉t ✶ ❋❧♦✇ P❛❝❦❡t s✐③❡ ❇✉rst ❚❤r♦✉❣❤♣✉t αi πi F1 ✶✴✷ ✶ ✶✴✹ ①✴✹ ✰✶ ⌈2x⌉ F2 ✶ ✶ ✶✴✹ ①✴✹ ✰✶ ⌈x⌉

• ♦❛❧✿ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ♣❛❝❦❡t t❤r♦✉❣❤♣✉t

F = F1 + F2 ✇❤❛t ✐s η ❄ ❝❤❛❧❧❡♥❣❡✿ ♠♦❞❡❧❧✐♥❣ t❤❡ ❧✐♥❦ s❤❛♣✐♥❣

✶✹✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❆❣❣r❡❣❛t✐♦♥

P❛❝❦❡t t❤r♦✉❣❤♣✉t✿ ♥♦ s❤❛♣✐♥❣

◆♦ s❤❛♣✐♥❣ ✿ η1 = π1 ◦ α1 = x

2

• + 2

η2 = π2 ◦ α2 = x

4

• + 1

η ≤ η1 + η2 η1 η2

✶✺✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❆❣❣r❡❣❛t✐♦♥

P❛❝❦❡t t❤r♦✉❣❤♣✉t✿ ✇✐t❤ s❤❛♣✐♥❣

▲✐♥❦ t❤r♦✉❣❤♣✉t✿ λ(t) = t ❙❤❛♣✐♥❣ r❡❞✉❝❡s ❞❛t❛ t❤r♦✉❣❤♣✉t

❢♦r ❡❛❝❤ ✢♦✇✱ αs

i = λ ∧ αi

❢♦r t❤❡ ❛❣❣r❡❣❛t❡ ✢♦✇✿ αs

1+2 = λ ∧ (α1 + α2)

■♠♣❛❝t ♦♥ ♣❛❝❦❡t t❤r♦✉❣❤♣✉t

♣❡r ✢♦✇✿ ηs

i = πi ◦ αs i

❛❣❣r❡❣❛t❡ ✢♦✇✿ ηs

1+2 = ⌈π1 ∗ π2⌉ ◦ αs 1+2

❜♦t❤ ηs

1 + ηs 2 ❛♥❞ ηs 1+2 ❛r❡ ♣❛❝❦❡t

t❤r♦✉❣❤♣✉t ❜♦✉♥❞s

α1 λ ∧ α1

(λ ∧ α1) + (id ∧ α2) λ ∧ (α1 + α2)

✶✻✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

◆❈✴❈P❆ ❘❡s✉❧ts ❆❣❣r❡❣❛t✐♦♥

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

t❤❡ s❤❛♣✐♥❣ ♦♥❧② ❛✛❡❝ts st❛rt ♦❢ ❝✉r✈❡ t❤❡ s✐♠♣❧❡ ♠❡t❤♦❞ ❤❛s ❜❡tt❡r ❧♦♥❣ t❡r♠ t❤r♦✉❣❤♣✉t t❤❡ ♥❡✇ ♠❡t❤♦❞ ✐s ❧♦❝❛❧❧② ❜❡tt❡r

✶✼✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ✸ ❘❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

◆❈✴❈P❆ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❆ st❡♣ ❢♦r✇❛r❞ ✐♥ ♠♦❞❡❧❧✐♥❣ ♣❛❝❦❡ts ❙♦♠❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts ❆❣❣r❡❣❛t✐♦♥ r❡s✉❧t st✐❧❧ ❞✐s❛♣♣♦✐♥t✐♥❣ ♦♥ r❡❛❧ ❡①❛♠♣❧❡s ▲❛r❣❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡✛♦rt

✶✽✴✶✽ ◆❈✴❈P❆ ❇♦②❡r✱ ❘♦✉①✱ ❖❝t✳ ✾t❤ ✷✵✷✵

❖✉t❧✐♥❡

✶ ❈✉r✈❡ ❜❛s❡❞ ♠♦❞❡❧s ✷ ❆ ❣❡♥❡r❛❧ ♠♦❞❡❧✱ ❛♥❞ ✐ts t♦♦❧s ✸ ❘❡s✉❧ts ✹ ❈♦♥❝❧✉s✐♦♥ ✺ ❇✐❜❧✐♦❣r❛♣❤②

❘❡❢❡r❡♥❝❡s

❬✶❪ ❆✳ ❇♦✉✐❧❧❛r❞✱ ▲✳ ❏♦✉❤❡t ❛♥❞ ❊✳ ❚❤✐❡rr②✱ ❙❡r✈✐❝❡ ❝✉r✈❡s ✐♥ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s✿ ❞♦s ❛♥❞ ❞♦♥✬ts✱ ❘❡s❡❛r❝❤ ❘❡♣♦rt ■◆❘■❆ ♥o ❘❘✲✼✵✾✹ ✭✷✵✵✾✮✳ ❬✷❪ ❆✳ ❇♦✉✐❧❧❛r❞✱ ▼✳ ❇♦②❡r ❛♥❞ ❊✳ ▲❡ ❈♦rr♦♥❝✱ ❉❡t❡r♠✐♥✐st✐❝ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s ✕ ❋r♦♠ t❤❡♦r② t♦ ♣r❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ ◆♦✳ ■❙❇◆✿ ✾✼✽✲✶✲✶✶✾✲✺✻✸✹✶✲✾✳ ❲✐❧❡② ✭✷✵✶✽✮✳ ❬✸❪ ▼✳ ❇♦②❡r ❛♥❞ P✳ ❘♦✉①✱ ❊♠❜❡❞❞✐♥❣ ♥❡t✇♦r❦ ❝❛❧❝✉❧✉s ❛♥❞ ❡✈❡♥t str❡❛♠ t❤❡♦r② ✐♥ ❛ ❝♦♠♠♦♥ ♠♦❞❡❧✱ ■♥ Pr♦❝✳ ♦❢ t❤❡ ✷✶st ■❊❊❊ ■♥t✳ ❈♦♥❢✳ ♦♥ ❊♠❡r❣✐♥❣ ❚❡❝❤♥♦❧♦❣✐❡s ❛♥❞ ❋❛❝t♦r② ❆✉t♦♠❛t✐♦♥ ✭❊❚❋❆ ✷✵✶✻✮✱ ❇❡r❧✐♥✱ ●❡r♠❛♥② ✭❙❡♣t❡♠❜❡r ✷✵✶✻✮✳ ❬✹❪ ▲✳ ❑ö❤❧❡✱ ❇✳ ◆✐❦♦❧✐➣ ❛♥❞ ▼✳ ❇♦②❡r✱ ■♥❝r❡❛s✐♥❣ ❆❝❝✉r❛❝② ♦❢ ❚✐♠✐♥❣ ▼♦❞❡❧s✿ ❋r♦♠ ❈P❆ t♦ ❈P❆✰✱ ■♥ Pr♦❝✳ ♦❢ t❤❡ ❉❡s✐❣♥✱ ❆✉t♦♠❛t✐♦♥ ❛♥❞ ❚❡st ✐♥ ❊✉r♦♣❡ ❈♦♥❢❡r❡♥❝❡ ❛♥❞ ❊①❤✐❜✐t✐♦♥ ✭❉❆❚❊✮✱ ❋❧♦r❡♥❝❡✱ ■t❛❧② ✭▼❛r❝❤ ✷✵✶✾✮✳

❬✺❪ ❆✳ ❇♦✉✐❧❧❛r❞✱ ◆✳ ❋❛r❤✐ ❛♥❞ ❇✳ ●❛✉❥❛❧✱ P❛❝❦❡t✐③❛t✐♦♥ ❛♥❞ P❛❝❦❡t ❈✉r✈❡s ✐♥ ◆❡t✇♦r❦ ❈❛❧❝✉❧✉s✱ ■♥ Pr♦❝✳ ♦❢ t❤❡ ✻t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ P❡r❢♦r♠❛♥❝❡ ❊✈❛❧✉❛t✐♦♥ ▼❡t❤♦❞♦❧♦❣✐❡s ❛♥❞ ❚♦♦❧s ✭❱❛❧✉❡❚♦♦❧s ✷✵✶✷✮✱ ❈❛r❣❡s❡✱ ❋r❛♥❝❡ ✭❖❝t♦❜❡r✱ ✾✕✶✷ ✷✵✶✷✮✱ ■♥✈✐t❡❞ P❛♣❡r✳ ❬✻❪ ▼✳ ❇♦②❡r ❛♥❞ P✳ ❘♦✉①✱ ❆ ❝♦♠♠♦♥ ❢r❛♠❡✇♦r❦ ❡♠❜❡❞❞✐♥❣ ♥❡t✇♦r❦ ❝❛❧❝✉❧✉s ❛♥❞ ❡✈❡♥t str❡❛♠ t❤❡♦r②✱ ✇♦r❦✐♥❣ ♣❛♣❡r ✴ ❤❛❧✲✵✶✸✶✶✺✵✷✳ ❬✼❪ ❆✳ ❇♦✉✐❧❧❛r❞✱ ◆✳ ❋❛r❤✐ ❛♥❞ ❇✳ ●❛✉❥❛❧✱ P❛❝❦❡t✐③❛t✐♦♥ ❛♥❞ ❆❣❣r❡❣❛t❡ ❙❝❤❡❞✉❧✐♥❣✱ ❚❡❝❤♥✐❝❛❧ r❡♣♦rt ■◆❘■❆ ♥o ✼✻✽✺ ✭✷✵✶✶✮✳