◗✉❡✉❡✐♥❣ ❙②st❡♠s ❙❤❛♥❦❛r ▼❛② ✼✱ ✷✵✶✺
◗✉❡✉❡✐♥❣ ❖✈❡r✈✐❡✇ ◗✉❡✉❡✐♥❣ s②st❡♠ s❡r✈❡rs ✰ ✇❛✐t✐♥❣ r♦♦♠s ❝✉st♦♠❡rs ❛rr✐✈❡✱ ✇❛✐t✱ ❣❡t s❡r✈❡❞✱ ❞❡♣❛rt ♦r ❣♦ t♦ ♥❡①t s❡r✈❡r q✉❡✉❡✐♥❣ ❞✐s❝✐♣❧✐♥❡s ♥♦♥✲♣r❡❡♠♣t✐✈❡✿ ✜❢♦✱ ♣r✐♦r✐t②✱ ✳✳✳ ♣r❡❡♠♣t✐✈❡✿ r♦✉♥❞✲r♦❜✐♥✱ ♠✉❧t✐✲❧❡✈❡❧ ❢❡❡❞❜❛❝❦✱ ✳✳✳ ❖♣❡r❛t✐♥❣ s②st❡♠s ❛r❡ ❡①❛♠♣❧❡s ♦❢ q✉❡✉❡✐♥❣ s②st❡♠s s❡r✈❡rs✿ ❤✇✴s✇ r❡s♦✉r❝❡s ✭♣r♦❝❡ss♦r✱ ❞✐s❦✱ s❝❤❡❞✉❧❡r✱ ✳✳✳✮ ❝✉st♦♠❡rs✿ P❈❇s✱ ❚❈❇s✱ ✳✳✳ ●✐✈❡♥✿ ❛rr✐✈❛❧ r❛t❡s✱ s❡r✈✐❝❡ t✐♠❡s✱ q✉❡✉❡✐♥❣ ❞✐s❝✐♣❧✐♥❡s✱ ✳✳✳ ❖❜t❛✐♥✿ q✉❡✉❡ s✐③❡s✱ r❡s♣♦♥s❡ t✐♠❡s✱ ❢❛✐r♥❡ss✱ ❜♦tt❧❡♥❡❝❦s✱ ✳✳✳
❇✉rst② ❚r❛✣❝✿ ❲❤② ◗✉❡✉❡s ❆r✐s❡ ❈♦♥s✐❞❡r ❝❛rs tr❛✈❡❧✐♥❣ ♦♥ ❛ r♦❛❞ ✇✐t❤ ❛ t✉r♥ ❡❛❝❤ ❝❛r t❛❦❡s ✸ s❡❝♦♥❞s t♦ ❣♦ t❤r♦✉❣❤ t❤❡ t✉r♥ ❛t ♠♦st ♦♥❡ ❝❛r ❝❛♥ ❜❡ ✐♥ t❤❡ t✉r♥ ❛t ❛♥② t✐♠❡ N ( t ) ✿ ★ ❝❛rs ✐♥ t❤❡ t✉r♥ ❛♥❞ ✇❛✐t✐♥❣ t♦ ❡♥t❡r t❤❡ t✉r♥ N(t) arrival rate 1/4 2 1 2 3 4 load 3/4 1 uniform t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 N(t) 1 3 2 arrival rate 1/4 2 4 load 3/4 1 bursty t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 ▲♦❛❞ < ✶✿ st❛❜❧❡ ✇✐t❤ ✇❛✐ts ❞❡♣❡♥❞✐♥❣ ♦♥ ❜✉rst✐♥❡ss ▲♦❛❞ > ✶✿ ✉♥st❛❜❧❡✱ ❡✈❡r✲✐♥❝r❡❛s✐♥❣ ✇❛✐ts
■♥st❛♥t❛♥❡♦✉s ▼❡tr✐❝s ❈✉st♦♠❡r i ♠❡tr✐❝s ❛rr✐✈❛❧ t✐♠❡ A i s❡r✈✐❝❡ t✐♠❡ S i ❞❡♣❛rt✉r❡ t✐♠❡ D i r❡s♣♦♥s❡ t✐♠❡ R i ✿ D i − A i ✇❛✐t t✐♠❡ W i ✿ R i − S i ◗✉❡✉❡ ♠❡tr✐❝s N ( t ) ✿ ★ ❝✉st♦♠❡rs ✐♥ q✉❡✉❡ ✭✇❛✐t✐♥❣ ✰ s❡r✈❡❞✮ ❛t t✐♠❡ t N W ( t ) ✿ ♥✉♠❜❡r ♦❢ ❝✉st♦♠❡rs ✐♥ ✇❛✐t✐♥❣ r♦♦♠ ❛t t✐♠❡ t Y ( t ) ✿ ✉♥✜♥✐s❤❡❞ ✇♦r❦ ✐♥ q✉❡✉❡ ✭✇❛✐t✐♥❣ ✰ s❡r✈❡❞✮ ❛t t✐♠❡ t
❙t❡❛❞②✲st❛t❡ ▼❡tr✐❝s ❆ss✉♠❡ ✉♥❡♥❞✐♥❣ str❡❛♠ ♦❢ ❝✉st♦♠❡rs ❈✉st♦♠❡r ♠❡tr✐❝s✱ ❛ss✉♠✐♥❣ N → ∞ ❛✈❣ s❡r✈✐❝❡ t✐♠❡ S ✿ ( S ✶ + · · · + S N ) / N ❛✈❣ r❡s♣♦♥s❡ t✐♠❡ R ✿ ( R ✶ + · · · + R N ) / N ✴✴ st❛❜❧❡ ❛✈❣ ✇❛✐t t✐♠❡ R ✿ ( W ✶ + · · · + W N ) / N ✴✴ st❛❜❧❡ ◗✉❡✉❡ ♠❡tr✐❝s✱ ❛ss✉♠✐♥❣ t → ∞ ❛rr✐✈❛❧ r❛t❡ λ ✿ ✭★ ❛rr✐✈❛❧s ✐♥ [ ✵ , t ] ✮ ✴ t ❧♦❛❞ ρ ✿ ✭t♦t❛❧ s❡r✈✐❝❡ t✐♠❡ ❛rr✐✈✐♥❣ ✐♥ [ ✵ , t ] ✮ ✴ t t❤r♦✉❣❤♣✉t X ✿ ✭★ ❞❡♣❛rt✉r❡s ✐♥ [ ✵ , t ] ✮ ✴ t ❛✈❣ q✉❡✉❡ s✐③❡ N ✿ ✭❛r❡❛ ✉♥❞❡r N ( t ) ♦✈❡r [ ✵ , t ] ✮ ✴ t ✴✴ st❛❜❧❡ ✉t✐❧✐③❛t✐♦♥ U ✿ ❢r❛❝t✐♦♥ ♦❢ [ ✵ , t ] ✐♥ ✇❤✐❝❤ s❡r✈❡r ✐s ❜✉s②
❙♦♠❡ ❙t❡❛❞②✲st❛t❡ ❘❡❧❛t✐♦♥s❤✐♣s ❋r♦♠ ❞❡✜♥✐t✐♦♥s ▲♦❛❞ ρ = λ × S ❙②st❡♠ ✐s ✉♥st❛❜❧❡ ✐❢ ρ > ✶ ❋♦r st❛❜❧❡ s②st❡♠✿ X = λ ❛♥❞ U = ρ ❋♦r ❛♥ ✉♥st❛❜❧❡ s②st❡♠✿ X = ✶ / S ❛♥❞ U = ✶ ▲✐tt❧❡✬s ▲❛✇ N = R × X ❛ss✉♠✐♥❣ st❡❛❞②✲st❛t❡ ❤♦❧❞s ❢♦r ❛♥② q✉❡✉❡✐♥❣ s②st❡♠✿ ❡❣✱ ❛ ❝❧❛ss ♦❢ ❝✉st♦♠❡rs
❙t❡❛❞②✲st❛t❡✿ ◗✉❡✉❡ ❙✐③❡ ✈s ▲♦❛❞ N ✐♥❝r❡❛s❡s ✏❡①♣♦♥❡♥t✐❛❧❧②✑ ❛s ρ ✐♥❝r❡❛s❡s✱ ❜❡❝♦♠✐♥❣ ∞ ❛s ρ → ✶ N ✐♥❝r❡❛s❡s ❛s ❜✉rst✐♥❡ss ✐♥❝r❡❛s❡s N increasing burstiness ρ 0 1.0
❙t❡❛❞②✲st❛t❡✿ ❲❛✐t t✐♠❡ ✈s ❙❡r✈✐❝❡ t✐♠❡ ◗✉❡✉✐♥❣ ❞✐s❝✐♣❧✐♥❡s ❝❛♥ ❞✐s❝r✐♠✐♥❛t❡ ❜❛s❡❞ ♦♥ s❡r✈✐❝❡ t✐♠❡s W ( S ) ✿ ❛✈❣ ✇❛✐t t✐♠❡ ❢♦r ❝✉st♦♠❡rs ✇✐t❤ s❡r✈✐❝❡ t✐♠❡ S ❋❛✈♦r ❝✉st♦♠❡rs ✇✐t❤ s♠❛❧❧ S ❙❏❋✲♣r❡❡♠♣t✐✈❡ > ❙❏❋ > ❘❘ > ❋■❋❖✱ ▲■❋❖ ❘❘ ✇ q✉❛♥t✉♠ → ✵✿ ❧✐♥❡❛r ❞✐s❝r✐♠✐♥❛t✐♦♥ ✴✴ ✐❣♥♦r✐♥❣ ♦✈❡r❤❡❛❞ W(S) RR (qs −−> 0) SJF FIFO S 0
❘❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ✐❞❧❡ ❛♥❞ ❜✉s② ♣❡r✐♦❞s ❙❡r✈❡r ❝②❝❧❡s ❜❡t✇❡❡♥ ✐❞❧❡ ♣❡r✐♦❞s ❛♥❞ ❜✉s② ♣❡r✐♦❞s ❲♦r❦✲❝♦♥s❡r✈✐♥❣ ❞✐s❝✐♣❧✐♥❡✿ s❡r✈❡r ♥♦t ✐❞❧❡ ✇❤❡♥ ❝✉st♦♠❡r ♣r❡s❡♥t Pr♦♣❡rt② ❢♦r ✇♦r❦✲❝♦♥s❡r✈✐♥❣ ❞✐s❝✐♣❧✐♥❡s✿ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ✐❞❧❡ ❛♥❞ ❜✉s② ♣❡r✐♦❞s✱ ❤❡♥❝❡ ✉t✐❧✐③❛t✐♦♥✱ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ q✉❡✉❡✐♥❣ ❞✐s❝✐♣❧✐♥❡✳ Pr♦♦❢ ✐s ♦❜✈✐♦✉s ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ Y ( t ) ❛rr✐✈❛❧ ✐♥❝r❡❛s❡s Y ( t ) ❜② ❛rr✐✈❛❧✬s s❡r✈✐❝❡ t✐♠❡ ✇❤✐❧❡ Y ( t ) > ✵ ❤♦❧❞s✱ ✐t ❞❡❝r❡❛s❡s ✇✐t❤ s❧♦♣❡ − ✶
❊✈♦❧✉t✐♦♥ ♦❢ ✉♥✜♥✐s❤❡❞ ✇♦r❦ Y ( t ) Y(t) unfinished 2 work 4 S 1 5 1 3 0 t 1 2 3 4 5
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