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❉✐str✐❜✉t❡❞ ❙♦❧✉t✐♦♥s ✉s✐♥❣ ❚✐♠❡st❛♠♣s ❙❤❛♥❦❛r ❆♣r✐❧ ✷✺✱ ✷✵✶✹

❊✈❡♥t ♦r❞❡r✐♥❣ ✐♥ ❛ ❞✐str✐❜✉t❡❞ s②st❡♠ sys 1 sys n fifo channel ▲❡t x ❛♥❞ y ❜❡ t✇♦ st❛t❡♠❡♥t ❡①❡❝✉t✐♦♥s ✭❛❦❛ ❡✈❡♥ts✮ ❉❡✜♥❡ x ❝❛✉s❛❧❧② ♣r❡❝❡❞❡s y ✐❢ x ❛♥❞ y ❤❛♣♣❡♥❡❞ ✐♥ t❤❛t ♦r❞❡r ✐♥ t❤❡ s❛♠❡ s②st❡♠✱ ♦r x s❡♥t ❛ ♠❡ss❛❣❡ t❤❛t y r❡❝❡✐✈❡❞✱ ♦r tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ♦❢ ❛❜♦✈❡ ❈❛✉s❛❧ ♣r❡❝❡❞❡♥❝❡ ✐s ❛ ♣❛rt✐❛❧ ♦r❞❡r ✐❢ x ❛♥❞ y ♥♦t ❝❛✉s❛❧❧② r❡❧❛t❡❞✱ ♥♦ s②st❡♠ ❝❛♥ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ❤❛♣♣❡♥❡❞ ✜rst ✭✇✐t❤♦✉t ♦t❤❡r ✐♥t❡r❛❝t✐♦♥ ♦r r❡❛❧✲t✐♠❡ ❝❧♦❝❦s✮ ❚✐♠❡st❛♠♣ ♠❡❝❤❛♥✐s♠ ❡①t❡♥❞s ♣❛rt✐❛❧ ♦r❞❡r t♦ t♦t❛❧ ♦r❞❡r ❢♦r ❛ s♣❡❝✐✜❡❞ s❡t ♦❢ ❡✈❡♥ts

❖✉t❧✐♥❡ ts ♠❡❝❤❛♥✐s♠ ❚✐♠❡st❛♠♣ ♠❡❝❤❛♥✐s♠ ❉✐str✐❜✉t❡❞ ♦r❞❡r✐♥❣ ♦❢ ❝♦♥✢✐❝t✐♥❣ r❡q✉❡sts ❉✐str✐❜✉t❡❞ ❧♦❝❦ ♣r♦❣r❛♠✿ ❛❧❣♦r✐t❤♠ ❧❡✈❡❧ ❉✐str✐❜✉t❡❞ ❧♦❝❦ ♣r♦❣r❛♠✿ ♦✈❡r✈✐❡✇ ❈②❝❧✐❝ t✐♠❡st❛♠♣s

❚✐♠❡st❛♠♣ ♠❡❝❤❛♥✐s♠ ✇✐t❤♦✉t ❛❝❦s ts ♠❡❝❤❛♥✐s♠ ❊❛❝❤ s②st❡♠ j ❤❛s ✐♥t❡❣❡r ✏❝❧♦❝❦✑ clk ✱ ✐♥✐t✐❛❧❧② ✵ ❲❤❡♥ j ❞♦❡s ❛♥ ❡✈❡♥t x t♦ ❜❡ ♦r❞❡r❡❞✿ ✐♥❝r❡♠❡♥t clk ✱ ❜r♦❛❞❝❛st [ x , clk , j ] clk value ✿ t✐♠❡st❛♠♣ ✭ts✮ ♦❢ x [ clk value , id ] ✿ ❡①t❡♥❞❡❞ t✐♠❡st❛♠♣ ✭❡ts✮ ♦❢ x ❲❤❡♥ j r❡❝❡✐✈❡s ♠s❣ [ y , t , k ] ✿ clk ← ♠❛① ( t , clk ) + ✶ x ✳❡ts < y ✳❡ts✿ ✭ x ✳ts < y ✳ts✮ ♦r ✭ x ✳ts = y ✳ts ❛♥❞ x ✳✐❞ < y ✳✐❞✮ ❊✈❡♥t x ♦r❞❡r❡❞ ❜❡❢♦r❡ ❡✈❡♥t y ✐❢ x ✳❡ts < y ✳❡ts ❋♦r ♠♦st ❛♣♣❧✐❝❛t✐♦♥s✱ ♥❡❡❞ ❛❝❦s t♦ t✐♠❡st❛♠♣s

❚✐♠❡st❛♠♣ ♠❡❝❤❛♥✐s♠ ✭✇✐t❤ ❛❝❦s✮ ts ♠❡❝❤❛♥✐s♠ ❊❛❝❤ s②st❡♠ j ❤❛s ✐♥t❡❣❡r ❝❧♦❝❦ clk ✱ ✐♥✐t✐❛❧❧② ✵ rts k ✱ k � = j ✱ ✐♥✐t✐❛❧❧② ✵ ✴✴ ❧❛st ts r❝✈❞ ❢r♦♠ k ❧❡t α Rts ✿ ♠✐♥ ([ rts k , k ] : k � = j ) ✴✴ ♥♦ ♥❡✇ ❡ts < α Rts ❲❤❡♥ j ❞♦❡s ❛♥ ❡✈❡♥t x t♦ ❜❡ ♦r❞❡r❡❞✿ ✐♥❝r❡♠❡♥t clk ✱ ❜r♦❛❞❝❛st [ x , clk , j ] ❲❤❡♥ j r❡❝❡✐✈❡s ♠s❣ [ y , t , k ] ✿ clk ← ♠❛① ( t , clk ) + ✶ rts k ← t ✱ s❡♥❞ [ ❛❝❦ , clk , j ] t♦ k ❲❤❡♥ j r❡❝❡✐✈❡s ♠s❣ [ ❛❝❦ , t , k ] ✿ clk ← ♠❛① ( t , clk ) ✱ rts k ← t

Pr♦♣❡rt✐❡s ts ♠❡❝❤❛♥✐s♠ ❉❡✜♥❡ ❛✉①✐❧✐❛r② q✉❛♥t✐t✐❡s hst ✿ s❡q ♦❢ ❛❧❧ ❡ts✬s ✐♥ ❡ts✲♦r❞❡r ✴✴ ✐♥✐t✐❛❧❧② [ [ ✵ , ✵ ] ] j . hst ✿ s❡q ♦❢ ❡ts✬s s❡❡♥ ❜② j ✐♥ ❡ts✲♦r❞❡r ✴✴ ✐♥✐t✐❛❧❧② [ [ ✵ , ✵ ] ] j .α hst ✿ ♣r❡✜① ♦❢ j . hst ♦❢ ❡ts✬s ≤ j .α Rts ❙❛❢❡t② ♣r♦♣❡rt✐❡s ■♥✈ j . hst s✉❜s❡q✉❡♥❝❡✲♦❢ hst j .α hst ♣r❡✜①✲♦❢ hst ■♥✈ Pr♦❣r❡ss ♣r♦♣❡rt✐❡s ✭❛ss✉♠✐♥❣ ♥♦ s②st❡♠ st♦♣s r❝✈✐♥❣✮ j .α Rts = z < hst . last ❧❡❛❞s✲t♦ j .α Rts > z j .α hst . size = z < hst . size ❧❡❛❞s✲t♦ j .α hst . size > z

❖✉t❧✐♥❡ r❡q✉❡st ♦r❞❡r✐♥❣ ❚✐♠❡st❛♠♣ ♠❡❝❤❛♥✐s♠ ❉✐str✐❜✉t❡❞ ♦r❞❡r✐♥❣ ♦❢ ❝♦♥✢✐❝t✐♥❣ r❡q✉❡sts ❉✐str✐❜✉t❡❞ ❧♦❝❦ ♣r♦❣r❛♠✿ ❛❧❣♦r✐t❤♠ ❧❡✈❡❧ ❉✐str✐❜✉t❡❞ ❧♦❝❦ ♣r♦❣r❛♠✿ ♦✈❡r✈✐❡✇ ❈②❝❧✐❝ t✐♠❡st❛♠♣s

❘❡q✉❡st ♦r❞❡r✐♥❣ ♣r♦❜❧❡♠ r❡q✉❡st ♦r❞❡r✐♥❣ ❈♦❧❧❡❝t✐♦♥ ♦❢ s②st❡♠s ❛tt❛❝❤❡❞ t♦ ❛ ✜❢♦ ❝❤❛♥♥❡❧ ❯s❡rs ✐ss✉❡ ❝♦♥✢✐❝t❛❜❧❡ r❡q✉❡sts t♦ t❤❡ s②st❡♠s ❊❛❝❤ s②st❡♠ s❤♦✉❧❞ s❡r✈❡ ✐ts r❡q✉❡sts s♦ t❤❛t ❝♦♥✢✐❝t✐♥❣ r❡q✉❡sts ❛r❡ ♥♦t s❡r✈❡❞ s✐♠✉❧t❛♥❡♦✉s❧② ✭❡✈❡♥ ❜② ❞✐✛❡r❡♥t s②st❡♠s✮ ❙♦♠❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t❤❡ ♣r♦❜❧❡♠ ❞✐str✐❜✉t❡❞ ❧♦❝❦ ❡✈❡r② t✇♦ r❡q✉❡sts ❝♦♥✢✐❝t ❞✐str✐❜✉t❡❞ r❡❛❞❡rs✲✇r✐t❡rs ❧♦❝❦ ❝❧❛ss✐❢② r❡q✉❡sts ✐♥t♦ r❡❛❞s ❛♥❞ ✇r✐t❡s ✇r✐t❡ ❝♦♥✢✐❝ts ✇✐t❤ ❡✈❡r② ♦t❤❡r r❡q✉❡st

❙♦❧✉t✐♦♥ ♦✈❡r✈✐❡✇ r❡q✉❡st ♦r❞❡r✐♥❣ ❙②st❡♠ j ❛✉❣♠❡♥ts t❤❡ ts ♠❡❝❤❛♥✐s♠ ❛s ❢♦❧❧♦✇s ▼❛✐♥t❛✐♥ ✈❛r✐❛❜❧❡ req ✿ s❡t ♦❢ ✏♦♥❣♦✐♥❣✑ r❡q✲❡ts t✉♣❧❡s ❯♣♦♥ ❧♦❝❛❧ r❡q✉❡st x ✿ ❛ss✐❣♥ ts✱ ❛❞❞ [x,ts,j] t♦ req ❯♣♦♥ r❝✈✐♥❣ [y,t,k] ✿ ♣r♦❝❡ss ts✱ ❛❞❞ [y,t,k] t♦ req ❙❡r✈❡ [x,t,j] ✐♥ req ✇❤❡♥✿ [t,j] < α Rts ❛♥❞ [t,j] < [u,k] ❢♦r ❡✈❡r② ❝♦♥✢✐❝t✐♥❣ [y,u,k] ✐♥ req ❆❢t❡r s❡r✈✐♥❣ [x,t,j] ✿ r❡♠♦✈❡ ✐t ❢r♦♠ req ✱ ❜❝❛st [REL,x,t,j] ❯♣♦♥ r❝✈✐♥❣ [REL,y,t,k] ✿ r❡♠♦✈❡ [y,t,k] ❢r♦♠ req

❙♦❧✉t✐♦♥✿ ✈❛r✐❛❜❧❡s ❛♥❞ ♠❡ss❛❣❡s r❡q✉❡st ♦r❞❡r✐♥❣ ❙②st❡♠ j ✈❛r✐❛❜❧❡s clk ✿ ✐♥✐t✐❛❧❧② 0 ✴✴ ❝❧♦❝❦ rtsk ✿ ✐♥✐t✐❛❧❧② 0 ✴✴ ❤✐❣❤❡st ts r❝✈❞ ❢r♦♠ k α Rts ✿ ♠✐♥ (rtsk, k]: k � = j) ✴✴ ♠✐♥ ❡ts ✐♥❞✉❝❡❞ ❜② rts req ✿ ✐♥✐t✐❛❧❧② ❡♠♣t② ✴✴ s❡t ♦❢ ♦✉tst❛♥❞✐♥❣ r❡q✉❡sts✲❡ts ▼❡ss❛❣❡s [REQ,x,t,k] ✴✴ r❡q✉❡st ♠s❣ [ACK,t,k] ✴✴ ❛❝❦ ♠s❣ [REL,x,t,k] ✴✴ r❡❧❡❛s❡ ♠s❣

❙♦❧✉t✐♦♥✿ s②st❡♠ j r✉❧❡s ✕ ✶ r❡q✉❡st ♦r❞❡r✐♥❣ ❯s❡r ✐s✉❡s r❡q✉❡st x clk + + s❡♥❞ [REQ,x,clk,j] t♦ ❡✈❡r② s②st❡♠ ❛❞❞ [x,clk,j] t♦ req ❘❡❝❡✐✈❡ [REQ,x,t,k] ✿ clk ← max(clk, t+1) rts[k] ← t s❡♥❞ [ACK,clk,j] t♦ k ✴✴ ♦♠✐t ✐❢ ❡ts > [t,k] ❛❧r❡❛❞② s❡♥t t♦ k ❛❞❞ [x,t,k] t♦ req ❙t❛rt s❡r✈✐♥❣ r❡q✉❡st [x,t,j] ✐♥ req ✇❤❡♥ [t,j] ≤ α Rts ❢♦r ❡✈❡r② [y,s,k] ✐♥ req st x ❝♦♥✢✐❝ts ✇✐t❤ y ✿ [t,j] ≤ [s,k]

❙♦❧✉t✐♦♥✿ s②st❡♠ j r✉❧❡s ✕ ✷ r❡q✉❡st ♦r❞❡r✐♥❣ ❋✐♥✐s❤ s❡r✈✐♥❣ r❡q✉❡st [x,t,j] ✿ r❡♠♦✈❡ [x,t,j] ❢r♦♠ req ❀ s❡♥❞ [REL,x,t,j] t♦ ❡✈❡r② ♦t❤❡r s②st❡♠✳ ❘❡❝❡✐✈❡ [ACK,t,k] ✿ clk ← max(clk, t); rts[k] ← t ❘❡❝❡✐✈❡ [REL,x,t,k] ✿ r❡♠♦✈❡ [x,t,k] ❢r♦♠ req ❛t♦♠✐❝✐t② ❛ss✉♠♣t✐♦♥✿ r✉❧❡s ❛r❡ ❛t♦♠✐❝ ♣r♦❣r❡ss ❛ss✉♠♣t✐♦♥✿ ✇❡❛❦ ❢❛✐r♥❡ss

❖✉t❧✐♥❡ ❞✐st ❧♦❝❦✿ ❛❧❣✲❧❡✈❡❧ ❚✐♠❡st❛♠♣ ♠❡❝❤❛♥✐s♠ ❉✐str✐❜✉t❡❞ ♦r❞❡r✐♥❣ ♦❢ ❝♦♥✢✐❝t✐♥❣ r❡q✉❡sts ❉✐str✐❜✉t❡❞ ❧♦❝❦ ♣r♦❣r❛♠✿ ❛❧❣♦r✐t❤♠ ❧❡✈❡❧ ❉✐str✐❜✉t❡❞ ❧♦❝❦ ♣r♦❣r❛♠✿ ♦✈❡r✈✐❡✇ ❈②❝❧✐❝ t✐♠❡st❛♠♣s