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❘❡♥❡✇❛❧ t❤❡♦r②

➽✇✐t❤ ✐♠♣r❡❝✐s✐♦♥s❄ ❉❛♥✐❡❧ ❑r♣❡❧✐❦1,2

1❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s✱

❉✉r❤❛♠ ❯♥✐✈❡rs✐t②

2❉❡♣❛rt♠❡♥t ♦❢ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱

❱➆❇ ✲ ❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❖str❛✈❛

❖✈✐❡❞♦✱ ❏✉❧② ✷✵✶✽

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶ ✴ ✷✵

✶ ❘❡❧✐❛❜✐❧✐t② t❤❡♦r② ✷ ❘❡♥❡✇❛❧ t❤❡♦r② ✸ ▼✉❧t✐✲st❛t❡ s②st❡♠s

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✷ ✴ ✷✵

❚❤❡ ❣♦❛❧

❲❡ ✇❛♥t t♦ ♠❛❦❡ t❤✐♥❣s ✇♦r❦ ❜✉t ✇❡ ❛r❡ ♥♦t s✉r❡ ❤♦✇ t♦ ❞♦ s♦✦ ❚❤❡ st❛t❡ ♦❢ s②st❡♠ ❞❡♣❡♥❞s ♦♥ ✐ts ❡♥✈✐r♦♥♠❡♥t✳ ❙♦♠❡ st❛t❡s ❛r❡ ❞❡s✐r❛❜❧❡✱ s♦♠❡ ❛r❡ ♥♦t ✲ t❤♦s❡ ✇❡ ❝❛❧❧ ❢❛✐❧✉r❡s✳ ❆ss❡ss♠❡♥t✿ ●✐✈❡♥ ✉♥❝❡rt❛✐♥ ♠♦❞❡❧s✱ ✇❤❛t ✐s t❤❡ ❝❤❛♥❝❡ t❤❛t ❛ s②st❡♠ ✇✐❧❧ ❢❛✐❧❄ ❉❡❝✐s✐♦♥ ♠❛❦✐♥❣✿ ●✐✈❡♥ ✉♥❝❡rt❛✐♥ ♠♦❞❡❧s✱ ✇❤❛t ✐s t❤❡ ❜❡st ♣♦ss✐❜❧❡ ❞❡s✐❣♥ ❝❤♦✐❝❡ t♦ ✏❡♥s✉r❡✑ s②st❡♠ ❢✉♥❝t✐♦♥❛❧✐t②❄ ✭❖r ❤♦✇ t♦ ♣r♦♣❡r❧② ❜❛❧❛♥❝❡ r❡❧✐❛❜✐❧✐t② ❛♥❞ ❡①♣❡♥s❡s❄✮

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✸ ✴ ✷✵

❆❡❣✐♥❣ ♠♦❞❡❧s

❋♦r ♠❛♥② s②st❡♠s ♦❢ ✐♥t❡r❡st✱ ❢❛✐❧✉r❡s ❛r❡ ♦❢t❡♥ ❞❡❧❛②❡❞✦ ❆ss✉♠❡ t❤❛t✿ ❙②st❡♠ st❛t❡ X ✐s ❜✐♥❛r② ✲ ❡✐t❤❡r ❢✉♥❝t✐♦♥❛❧ ♦r ❢❛✐❧❡❞ ✭X ∈ {1, 0}✮✳ ❙②st❡♠ ✐s ❢✉♥❝t✐♦♥❛❧ ❛t t✐♠❡ ✵✳ ❖♥❝❡ s②st❡♠ ❢❛✐❧s✱ ✐t r❡♠❛✐♥s ✐♥ ❢❛✐❧❡❞ st❛t❡✳ ❚❤❡♥✱ t❤❡ s②st❡♠ st❛t❡ ♣r♦❝❡ss X(t) ❝❛♥ ❜❡ ❡q✉✐✈❛❧❡♥t❧② ❞❡s❝r✐❜❡❞ ❜② ❛ ❘❱ T > 0✱ r❡♣r❡s❡♥t✐♥❣ t❤❡ t✐♠❡ t♦ ❢❛✐❧✉r❡✱ ❛♥❞ Pr(X(t) = 1) = Pr(T > t)✳

• ✐✈❡♥ s♦♠❡ ❝♦♥t✐♥✉✐t② ❛ss✉♠♣t✐♦♥s✱ T ❝❛♥ ❜❡ ✉♥✐q✉❡❧② ❞❡s❝r✐❜❡❞ ❜②

❡✐t❤❡r ♦❢ F, R, f, r : R≥0 → R≥0✱ t❤❡ ❞✐str✐❜✉t✐♦♥✱ s✉r✈✐✈❛❧✱ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t②✱ ❛♥❞ ❤❛③❛r❞ r❛t❡ ❢✉♥❝t✐♦♥s✱ r❡s♣❡❝t✐✈❡❧②✳ ❉✐str✐❜✉t✐♦♥ ♦❢ T ❝❛♥ ❜❡ ✐♥❢❡rr❡❞ ❢r♦♠ ♦❜s❡r✈❛t✐♦♥s ♦❢ ❢❛✐❧✉r❡s✳

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✹ ✴ ✷✵

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❢❛✐❧✉r❡ ❧❛✇s ❬❇❛r❧♦✇ ❛♥❞ Pr♦s❝❤❛♥ ✶✾✻✼❪

❙❝✐❡♥t✐sts ❧♦✈❡ t❤❡✐r ③♦♦s✦ ❚❤❡ ❤❛③❛r❞ r❛t❡ ❢✉♥❝t✐♦♥ r(t) := lim∆→0

Pr(T∈[t;t+∆]|T>t) ∆

✐s ❡✛❡❝t✐✈❡❧② t❤❡ tr❛♥s✐t✐♦♥ r❛t❡ ❢r♦♠ ❢✉♥❝t✐♦♥❛❧ t♦ ❢❛✐❧❡❞ st❛t❡✳ ▼❛♥② ✉s❡❢✉❧ ♣r♦♣❡rt✐❡s ♠❛② ❜❡ ❞❡r✐✈❡❞ ❢♦r ❢❛✐❧✉r❡ ❧❛✇s ❜❡❧♦♥❣✐♥❣ t♦ s♣❡❝✐✜❝ ❝❧❛ss❡s✳ ❚ ✐s s❛✐❞ t♦ ❜❡ ♦❢✿ ■♥❝r❡❛s✐♥❣ ❋❛✐❧✉r❡ ❘❛t❡ ✐❢ r(t) ✐s ✐♥❝r❡❛s✐♥❣✳ ■♥❝r❡❛s✐♥❣ ❋❛✐❧✉r❡ ❘❛t❡ ✐♥ ❆✈❡r❛❣❡ ✐❢ t

0 r(x)dx/t ✐s ✐♥❝r❡❛s✐♥❣✳

◆❡✇ ❇❡tt❡r t❤❛♥ ❯s❡❞ ✐❢ R(x|t) ≤ R(x); ∀x ≥ t ≥ 0✳ ✳✳✳

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✺ ✴ ✷✵

■♠♣r❡❝✐s❡ r❡❧✐❛❜✐❧✐t② ❬▲❛✐ ❛♥❞ ❳✐❡ ✷✵✵✻❪

❊♥❣✐♥❡❡rs ❤❛✈❡ ❜❡❡♥ ❞♦✐♥❣ ■P s✐♥❝❡ ❛t ❧❡❛st t❤❡ ✻✵s✦ ◗✉❛❧✐t❛t✐✈❡ ❛ss❡ss♠❡♥ts ♠❛② ❜❡ ♠❛❞❡✱ ✇❤✐❝❤ ❤❡❧♣s ✉s t♦ ❝♦♥str✉❝t ❜♦✉♥❞s ♦♥ t❤❡ s✉r✈✐✈❛❧ ❢✉♥❝t✐♦♥ ✭Pr(T ≥ t)✮✳ ■❢ T ✐s ■❋❘ ❛♥❞ µr ✐ts r✲t❤ ❣❡♥❡r❛❧ ♠♦♠❡♥t✱ t❤❡♥✿ R(t) ≥ exp(−αt) ∀t < µ1/r

r

, α = [Γ(r + 1)/µr]1/r. ■❢ T ✐s ■❋❘❆ ❛♥❞ ξp ✐ts p−t❤ q✉❛♥t✐❧❡ ❛♥❞ α t❤❡ r❛t❡ ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ t❤❡ s❛♠❡ q✉❛♥t✐❧❡✱ t❤❡♥✿ R(t)

• ≥ exp(−αt)

0 ≤ t ≤ ξp, ≤ exp(−αt) t > ξp. ■❢ T ✉s ◆❇❯ s✳t✳ R(x) = α✱ t❤❡♥✿ R(t)

• ≥ α1/k

x k+1 ≤ t ≤ x k, k ∈ N,

≤ αk kx ≤ t ≤ (k + 1)x, k ∈ N.

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✻ ✴ ✷✵

❈♦♠♣❧❡① s②st❡♠s

❙②st❡♠s ❛r❡ s②st❡♠s ♦❢ s②st❡♠s✦ ❙②st❡♠s✱ ✇❡ ✉s❡✱ ❛r❡ ❝♦♠♣♦s❡❞ ♦❢ s✉❜✲s②st❡♠s✱ st❛t❡s ♦❢ ✇❤✐❝❤ ✐♥✢✉❡♥❝❡ t❤❡ st❛t❡ ♦❢ t❤❡ s✉♣❡r✲s②st❡♠✳ ❚❤✐s ❞❡♣❡♥❞❡♥❝② ✐s ♠♦❞❡❧❧❡❞ ❜② ❛ r❡❧✐❛❜✐❧✐t② ❢✉♥❝t✐♦♥ h : {p1, . . . , pn} → [0, 1]✱ ✇❤✐❝❤ ✐s ♠♦♥♦t♦♥❡ ✭❢♦r r❡❛s♦♥❛❜❧❡ s②st❡♠s✮✳ ⇒ ❲❡ ❝❛♥ ✐♥❢❡r t❤❡ ❞✐str✐❜✉t✐♦♥ ❧❛✇s ❢♦r t❤❡ s✉❜✲s②st❡♠s ✭❝❤❡❛♣❡r✮ ❛♥❞ t❤❡ ❞❡♣❡♥❞❡♥❝② ♠♦❞❡❧ t♦ ❛ss❡ss r❡❧✐❛❜✐❧✐t② ♦❢ t❤❡ s✉♣❡r✲s②st❡♠ ✭❛❧s♦ ❝❤❡❛♣❡r t❤❛♥ ❜r❡❛❦✐♥❣ t❤❡ s②st❡♠ ♦✈❡r ❛♥❞ ♦✈❡r ❛❣❛✐♥✮✳

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✼ ✴ ✷✵

▼❛✐♥t❡♥❛♥❝❡

■❢ ❛ s②st❡♠ ❢❛✐❧s✱ ✇❡ ❝❛♥ r❡♣❛✐r ✐t✱ ♦r ❜✉② ❛ ♥❡✇ ♦♥❡✦ ❚❤❡r❡ ❡①✐sts ♠❛♥② ♠❛✐♥t❡♥❛♥❝❡ s❝❡♥❛r✐♦s✳ ❈♦rr❡❝t✐✈❡ ♠❛✐♥t❡♥❛♥❝❡ ✭r❡♣❧❛❝❡♠❡♥t ✉♣♦♥ ❞✐s❝♦✈❡r② ♦❢ ❢❛✐❧✉r❡✮ Pr❡✈❡♥t✐✈❡ ♠❛✐♥t❡♥❛♥❝❡ ✭r❡♣❧❛❝❡♠❡♥t ✉♣♦♥ ❢❛✐❧✉r❡ ❛♥❞ ❛t s♣❡❝✐✜❡❞ t✐♠❡✮ P❛rt✐❛❧ ♠❛✐♥t❡♥❛♥❝❡ ✭s②st❡♠ ✐s ❥✉st ♠❛❞❡ ♦♣❡r❛t✐♦♥❛❧✱ ♥♦t ❛s ❣♦♦❞ ❛s ♥❡✇✮ ❇♦t❤ ❢♦r✿ ❙②st❡♠s ✇✐t❤ ✐♠♠❡❞✐❛t❡ r❡♣❛✐r ✲ r❡♥❡✇❛❧ ♣r♦❝❡ss ❙②st❡♠s ✇✐t❤ ♣♦s✐t✐✈❡ r❡♣❛✐r t✐♠❡ ✲ ❛❧t❡r♥❛t✐♥❣ ♣r♦❝❡ss ❙②st❡♠s ✇✐t❤ ❧❛t❡♥t ❢❛✐❧✉r❡s ✭❡✳❣✳ ✇❡ ❝❛♥ ♦❜s❡r✈❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ♦♥❧② ❛t ✏✐♥s♣❡❝t✐♦♥ t✐♠❡s✑✮ ❙②st❡♠s ✇✐t❤ ✜♥✐t❡ ❛♠♦✉♥t ♦❢ s❡r✈✐❝❡ ♣❡rs♦♥❛❧ ❛♥❞✴♦r ❧✐♠✐t❡❞ r❡s♦✉r❝❡s ✳ ✳ ✳

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✽ ✴ ✷✵

❘❡♥❡✇❛❧ ♣r♦❝❡ss ✲ ❞❡✜♥✐t✐♦♥

▲❡t ✉s ❛ss✉♠❡ t❤❛t ❛ s②st❡♠ r❡♣❧❛❝❡❞ ❜② ❛ ♥❡✇ ♦♥❡ ✐♠♠❡❞✐❛t❡❧② ✉♣♦♥ ✐ts ❢❛✐❧✉r❡✳ ❲❡ ❝❛♥ ♠♦❞❡❧ t❤❡ r❡♣❧❛❝❡♠❡♥t t✐♠❡s ❛s ❛ ♣♦✐♥t ♣r♦❝❡ss✿ ❇❡ {Ti; i ∈ N} ♣♦s✐t✐✈❡ ❘❱s ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❛✇s Fi, Ri, fi, ri✳ ❉❡✜♥❡ S0 = 0 ❛♥❞ Sn := n

k=1 Tk ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ❧❛✇s

F n, Rn, fn, rn✱ ⇒

F n = F ∗ F n−1 ❛♥❞ f n = f ∗ f n−1

❚❤❡ P♦✐♥t Pr♦❝❡ss ♦❢ ✐♥t❡r❡st ✐s ∞

n=1 δSn✳

❋✐❣✉r❡✿ ❆ ♣❛rt ♦❢ ❛ ♣♦✐♥t ♣r♦❝❡ss✳

❚❤❡ r❡♥❡✇❛❧ ♣r♦❝❡ss {N(t) ∈ Z≥0; t ≥ 0} ♠♦❞❡❧s t❤❡ ♥✉♠❜❡r ♦❢ r❡♥❡✇❛❧s ✐♥ ✐♥t❡r✈❛❧ (0, t]✱ ✐✳❡✳✿ Pr(N(t) ≥ n) = Pr(Sn ≤ t)

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✾ ✴ ✷✵

❘❡♥❡✇❛❧ ♣r♦❝❡ss ✲ ♣r♦♣❡rt✐❡s

❖✈❡r t❤❡ ②❡❛rs✱ t❤❡ ✐♥t❡r❡st ❤❛s ❜❡❡♥ ♣✉t ♦♥ ❡✈❛❧✉❛t✐♥❣ t❤❡ ♠❡❛♥ ♦❢ r❡♥❡✇❛❧ ♣r♦❝❡ss ✭❢♦r ❡①♣❡❝t❡❞ ✉t✐❧✐t② ♣✉r♣♦s❡s✮✳ ❉❡♥♦t❡ M(t) := E{N(t)} ✭❛❦❛ t❤❡ r❡♥❡✇❛❧ ❢✉♥❝t✐♦♥✮✳ ❚❤❡♥✿ N(t) ✐s s❡♠✐✲▼❛r❦♦✈ ♣r♦❝❡ss ✭▼❛r❦♦✈ ❢♦r ❝♦♥st❛♥t ❢❛✐❧✉r❡ r❛t❡✮ (N(t), SN(t)) ✐s ▼❛r❦♦✈ ♣r♦❝❡ss ♦♥ Z ⊗ R M(t) = ∞

n=1 F n(t) ✭❜② ❞❡✜♥✐t✐♦♥✮✳

M(t) = F(t) + (F ∗ M)(t) ✭r❡♥❡✇❛❧ ❡q✉❛t✐♦♥✮ M(t) = M(Sn) + M(t − Sn) ✭✐❢ Ti ❛r❡ ✐✳✐✳❞✱ r❡st❛rt✐♥❣ ♣r♦♣❡rt②✮ limt→∞

M(t) t

=

1 E{T1} ✭✐❢ Ti ❛r❡ ✐✳✐✳❞✱ ❘❡♥❡✇❛❧ t❤❡♦r❡♠✮

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✵ ✴ ✷✵

❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ r❡♥❡✇❛❧ ❢✉♥❝t✐♦♥ ❬❖s❛❦✐ ✷✵✵✷❪

❋♦r s♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ Ti ✐✳✐✳❞✿ ❊①♣♦♥❡♥t✐❛❧ ✇✐t❤ r❛t❡ λ✿ M(t) = λt✳ P❤❛s❡✲t②♣❡ ❞✐str✐❜✉t❡❞✿ M(t) = t−v(t)T −1e−1

E[T]

,

dv dt (t) = v(t)Q∗✳

❋♦r s♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ Ti✱ M(t) ♠❛② ❜❡ ❜♦✉♥❞❡❞✱ ❡✳❣✳ ❢♦r Ti ✐✳✐✳❞✳✿ ◆❇❯❊✿

t E[T] − 1 ≤ M(t) ≤ t E[T]✳

■❋❘✿

t E[T] ≤ M(t) ≤ tF(t) E[T] ✳

❖t❤❡r✇✐s❡✱ t❤❡r❡ ✐s ♥♦ ❝❧♦s❡❞ s♦❧✉t✐♦♥ ❛♥❞ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ❤❛✈❡ t♦ ❜❡ ❡♠♣❧♦②❡❞✿ ▲❛♣❧❛❝❡ ✐♥✈❡rs✐♦♥✱ ❙♣❧✐♥❡ ❛♣♣r♦①✐♠❛t✐♦♥✱ ❘❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥✳

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✶ ✴ ✷✵

❘❡♥❡✇❛❧ r❡✇❛r❞ ♣r♦❝❡ss

❚❛❦❡✱ ❛❣❛✐♥✱ t❤❡ ♣♦✐♥t ♣r♦❝❡ss {Sn; n ∈ Z≥0}✳ ❆ss✐❣♥ ✇✐t❤ ❡❛❝❤ ❥✉♠♣ t✐♠❡ Sn ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ Rn✱ t❤❡ r❡✇❛r❞✴❝♦st✳ ✭❖❜s❡r✈❡ t❤❛t ❢♦r r❡♥❡✇❛❧ ♣r♦❝❡ss Rn = 1 ✮✳ ❉❡✜♥❡ R(t) = N(t)

i=1 Rn✳

❚❤❡♥✿ R(t) ✐s s❡♠✐✲▼❛r❦♦✈✳ (R(t), SN(t)) ✐s ▼❛r❦♦✈✳ limt→∞

R(t) t

= E{R1}

E{T1} ✭✐❢ Ti, Ri ❛r❡ ✐✳✐✳❞✱ ❘❡♥❡✇❛❧ r❡✇❛r❞

t❤❡♦r❡♠✮

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✷ ✴ ✷✵

❊①❛♠♣❧❡ ✲ ♦♣t✐♠❛❧ ♣r❡✈❡♥t✐✈❡ ♠❛✐♥t❡♥❛♥❝❡

❲❡ r❡♣❧❛❝❡ ❝♦♠♣♦♥❡♥t ✉♣♦♥ ❢❛✐❧✉r❡ ✭❝♦rr❡❝t✐✈❡ ♠❛✐♥t❡♥❛♥❝❡✮ ❖❘ ❛❢t❡r ♣r❡❞❡✜♥❡❞ t✐♠❡ T ∗ ✭♣r❡✈❡♥t✐✈❡ r❡♣❧❛❝❡♠❡♥t✮✳ Pr❡✈❡♥t✐✈❡ r❡♣❧❛❝❡♠❡♥t ❝♦sts c1 ❛♥❞ ❝♦rr❡❝t✐✈❡ r❡♣❧❛❝❡♠❡♥t ❝♦sts c2 > c1✳ ❚❤❡♥ ♠❡❛♥ ❝♦st ♣❡r ❝②❝❧❡ ✭✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ t✇♦ r❡♣❧❛❝❡♠❡♥ts✮ ✐s✿ c2Pr(T ≤ T ∗) + c1Pr(T > T ∗) T ∗ R(t)dt . ❲❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ T ∗❄ ❋♦r T ✇✐t❤ ❞❡❝r❡❛s✐♥❣ ❢❛✐❧✉r❡ r❛t❡ ✭◆❡✇ ❲♦rs❡ t❤❛♥ ❯s❡❞✮✱ T ∗ = ∞✳

• ❡♥❡r❛❧❧②✱ t❤❡ s♦❧✉t✐♦♥ s❛t✐s✜❡s✿

r(T ∗) T ∗ R(t)dt − F(T ∗) = c1 c2 − c1 . ❲❡ ❝❛♥ ❞♦ ✐t ❡❛s✐❡r ✇✐t❤ ■P✦✦✦

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✸ ✴ ✷✵

❆❧t❡r♥❛t✐♥❣ ♣r♦❝❡ss

❇❡✿ Tn, Gn; n ∈ N r❡s♣❡❝t✐✈❡❧② ✐✳✐✳❞✳ ❘❱s✳ ✭t✐♠❡ t♦ ❢❛✐❧✉r❡✱ t✐♠❡ t♦ r❡♣❛✐r✱ r❡s♣❡❝t✐✈❡❧②✮ Zn := Tn + Gn✱ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ n✲t❤ ❝②❝❧❡✳ Sn := Zn−1 + Tn✱ t❤❡ t✐♠❡ t♦ n✲t❤ ❢❛✐❧✉r❡✳ {X(t); t ∈ R≥0} ❛ ♣r♦❝❡ss ♦❢ s②st❡♠ st❛t❡ s✳t✳ X(t) :=

• ; ∃n : Sn ≤ t < Zn

1 ; else A(t) := E{X(t)}✱ t❤❡ ❛✈❛✐❧❛❜✐❧✐t②✳ ❚❤❡♥✿ lim

t→∞ A(t) = E{T1}

E{Z1}.

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✹ ✴ ✷✵

❆❧t❡r♥❛t✐♥❣ ♣r♦❝❡ss ❝♦♠♣✉t❛t✐♦♥

■❢ T, G ❛r❡ ❡①♣♦♥❡♥t✐❛❧ ✇✐t❤ r❛t❡s λ, µ r❡s♣❡❝t✐✈❡❧②✱ t❤❡♥ A(t) =

λ λ+µ (1 + exp (−(λ + µ)t))✳

• ❡♥❡r❛❧❧②✱ U(t) := 1 − A(t) s❛t✐s✜❡s r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛✿

U(t) = t fT (x)[1 − FG(t − x)]dx + t (fT ∗ fG)(x)U(t − x)dx. ❊✈❡♥ ♠♦r❡ ❣❡♥❡r❛❧❧②✿ ❙✐♠✉❧❛t❡✳

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✺ ✴ ✷✵

▼✉❧t✐✲st❛t❡ s②st❡♠s

❙♦♠❡t✐♠❡s ✐t ✐s ❞❡s✐r❛❜❧❡ t♦ ❞✐st✐♥❣✉✐s❤ st❛t❡s ♦❢ ♣❛rt✐❛❧ ❢❛✐❧✉r❡✳ ❇❡ X ✐♥ {0, 1, . . . , K}✱ s✳t✳ 0 r❡♣r❡s❡♥ts ❢❛✐❧✉r❡ st❛t❡✱ K ❢✉❧❧② ❢✉♥❝t✐♦♥❛❧ ♦♥❡ ❛♥❞ t❤❡ r❡st ♣❛rt✐❛❧ ❢❛✐❧✉r❡s✳ ❲❡ t❤❡♥ ❤❛✈❡ t♦ ♠♦❞❡❧ ♠♦r❡ ❣❡♥❡r❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss X(t)✳

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✻ ✴ ✷✵

❉②♥❛♠✐❝ r❡❧✐❛❜✐❧✐t②

❋✐❣✉r❡✿ P✐❡❝❡✇✐s❡ ❞❡t❡r♠✐♥✐st✐❝ ▼❛r❦♦✈ ♣r♦❝❡ss❀ ✐♠❣✳ ❢r♦♠ ❬▲❛❜❡❛✉✱ ❙♠✐❞ts✱ ❛♥❞ ❙✇❛♠✐♥❛t❤❛♥ ✷✵✵✵❪

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✼ ✴ ✷✵

❈♦♥❝❧✉s✐♦♥s

■ s❤♦✉❧❞ ❤❛✈❡ st✐❝❦ ✇✐t❤ ❜❡✐♥❣ ❛♥ ❡❧❡❝tr✐❝✐❛♥ ✲ ❧✐❢❡ ✇♦✉❧❞ ❤❛✈❡ ❜❡❡♥ ♠✉❝❤ ❡❛s✐❡r ♥♦✇✳✳✳ ❇✉t ♠✉❝❤ ❧❡ss ✐♥t❡r❡st✐♥❣ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✦ ❆ ❧♦t ♦❢ ✭❝♦❤❡r❡♥t❄✮ ❜♦✉♥❞s ❤❛s ❜❡❡♥ ❞✐s❝♦✈❡r❡❞ ✲ ♣♦ss✐❜❧② ❡❛s✐❧② ❡①t❡♥❞❡❞ ❢♦r ✐♠♣r❡❝✐s❡ ❛ss❡ss♠❡♥ts✳ ❘❡st❛rt✐♥❣ ♣r♦♣❡rt② ❧♦♦❦s ❧✐❦❡ ✇♦rt❤ ❡①♣❧♦✐t✐♥❣✳ ❍♦✇ ❞♦❡s ✐t ❧♦♦❦ ✐♥ ▲❛♣❧❛❝❡✬s ✇♦r❧❞❄ ❈❛♥ ✇❡ ✜♥❞ ❞♦♠✐♥❛t✐♥❣ ♣r♦❝❡ss❡s✱ ✇❤✐❝❤ ❛r❡ t✐❣❤t ❡♥♦✉❣❤ t♦ ♠❛❦❡ r❡❛s♦♥❛❜❧❡ ❛ss❡ss♠❡♥ts❄ ❈❛♥ ✇❡ s✐♠✉❧❛t❡ ✇✐t❤ ✐♠♣r❡❝✐s❡❧② s♣❡❝✐✜❡❞ ❧❛✇s❄❄✦❄✦❄✦❄❄✦❄❄✦❄✦

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✽ ✴ ✷✵

❚❤❛♥❦ ❨♦✉ ❢♦r ❨♦✉r ❛tt❡♥t✐♦♥✦

❇② t❤❡ ✇❛②✳✳✳ ❚❤✐s ✇♦r❦ ✐s ❢✉♥❞❡❞ ❜② t❤❡ ❊✉r♦♣❡❛♥ ❈♦♠♠✐ss✐♦♥✬s ❍✷✵✷✵ ♣r♦❣r❛♠♠❡✱ t❤r♦✉❣❤ t❤❡ ❯❚❖P■❆❊ ▼❛r✐❡ ❈✉r✐❡ ■♥♥♦✈❛t✐✈❡ ❚r❛✐♥✐♥❣ ◆❡t✇♦r❦✱ ❍✷✵✷✵✲▼❙❈❆✲■❚◆✲✷✵✶✻✱ ●r❛♥t ❆❣r❡❡♠❡♥t ♥✉♠❜❡r ✼✷✷✼✸✹✳

✭②♦✉ ❝❛♥ ❢♦❧❧♦✇ ✉s ♦♥ ❅❘❡s❡❛r❝❤●❛t❡✿♣r♦❥❡❝t ❯❚❖P■❆❊✮

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✶✾ ✴ ✷✵

❘❡❢❡r❡♥❝❡s

❘✐❝❤❛r❞ ❊✳ ❇❛r❧♦✇ ❛♥❞ ❋r❛♥❦ Pr♦s❝❤❛♥✳ ▼❛t❤❡♠❛t✐❝❛❧ t❤❡♦r② ♦❢ r❡❧✐❛❜✐❧✐t② ✴ ❘✐❝❤❛r❞ ❊✳ ❇❛r❧♦✇✱ ❋r❛♥❦ Pr♦s❝❤❛♥✱ ✇✐t❤ ❝♦♥tr✐❜✉t✐♦♥s ❜② ▲❛rr② ❈✳ ❍✉♥t❡r✳ ❊♥❣❧✐s❤✳ ❏♦❤♥ ❲✐❧❡② ◆❡✇ ❨♦r❦✱ ✶✾✻✼✳ P✳❊✳ ▲❛❜❡❛✉✱ ❈✳ ❙♠✐❞ts✱ ❛♥❞ ❙✳ ❙✇❛♠✐♥❛t❤❛♥✳ ✏❉②♥❛♠✐❝ r❡❧✐❛❜✐❧✐t②✿ t♦✇❛r❞s ❛♥ ✐♥t❡❣r❛t❡❞ ♣❧❛t❢♦r♠ ❢♦r ♣r♦❜❛❜✐❧✐st✐❝ r✐s❦ ❛ss❡ss♠❡♥t✑✳ ■♥✿ ❘❡❧✐❛❜✐❧✐t② ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❙②st❡♠ ❙❛❢❡t② ✻✽✳✸ ✭✷✵✵✵✮✱ ♣♣✳ ✷✶✾✕✷✺✹✳ ❈❤✐♥✲❉✐❡✇ ▲❛✐ ❛♥❞ ▼✐♥ ❳✐❡✳ ✏❈♦♥❝❡♣ts ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❙t♦❝❤❛st✐❝ ❆❣❡✐♥❣✑✳ ■♥✿ ❙t♦❝❤❛st✐❝ ❆❣❡✐♥❣ ❛♥❞ ❉❡♣❡♥❞❡♥❝❡ ❢♦r ❘❡❧✐❛❜✐❧✐t②✳ ◆❡✇ ❨♦r❦✱ ◆❨✿ ❙♣r✐♥❣❡r ◆❡✇ ❨♦r❦✱ ✷✵✵✻✳ ❈❤❛♣✳ ✷✱ ♣♣✳ ✼✕✼✵✳ ✐s❜♥✿ ✾✼✽✲✵✲✸✽✼✲✸✹✷✸✷✲✵✳ ❙❤✉♥❥✐ ❖s❛❦✐✱ ❡❞✳ ❙t♦❝❤❛st✐❝ ▼♦❞❡❧s ✐♥ ❘❡❧✐❛❜✐❧✐t② ❛♥❞ ▼❛✐♥t❡♥❛♥❝❡✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✷✳ ✐s❜♥✿ ✾✼✽✲✸✲✺✹✵✲✹✸✶✸✸✲✻✳

❉❛♥❞❛ ✭❉✉r❤❛♠✱ ❱➆❇✮ ❘❡♥❡✇❛❧ t❤❡♦r② ❖✈✐❡❞♦ ✷✵✶✽ ✷✵ ✴ ✷✵