D. Bertsekas and R. Gallager, "Data networks." Q: What are the labels for the x-axis and y-axis of Fig. 4.2?

Transcription

1 pd by J. Succ ECE 543 Octob Outl Slottd Aloh Dft Stblzd Slottd Aloh Uslottd Aloh Splttg Algoths Rfc D. Btsks d R. llg "Dt twoks." Rvw (Slottd Aloh): : Wht th lbls fo th x-xs d y-xs of Fg. 4.2? : Wht s th ky ssupto tht ylds th Fg. 4.2? cuv of : Fo wht typ of tffc ptt ght ths ssupto b lstc? : Wht s th xu thotcl thoughput of slottd loh? : Wht s fo xu thoughput?

2 pd by J. Succ ECE 543 Octob Dft D Expctd chg bcklog ov o t slot gv bckloggd pckts. D ( ) succss (4.4) Rcllg (4.) d (4.2): (4.) (4.2) ( ) ( ) ( ) ( ) succss y b xpssd s follows: succss ( ) ( 0 ) ( 0 ) ( ) (4.5) Futh t s vdt tht th tssso ttpt t () dpds o th ub of bckloggd pckts : ( ) ( ) Assug d ~ sll: ( ) ( ) succss (4.6) How s (4.6) dvd? 2

3 pd by J. Succ ECE 543 Octob ( ) ( ) succss??? (4.6) ( ) ( ) ( ) ( ) 0 0 succss (4.5) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) succss ( ) ( ) ( ) [ ] ( ) ( ) succss ( ) ( ) ( ) ( ) fo d ~ sll ( ) [ ] ( ) ( ) succss Rcllg tht ( ) ( ) ( ) ( ) ( ) succss Applyg ( ) y x y x sll x: ( ) ( ) d ( ) ( ) ( ) ( ) ( ) succss ( ) ( ) succss

4 pd by J. Succ ECE 543 Octob Applyg (4.6) to (4.4) d cosdg Fg. 4.4: D ( ) ( ) ( ) ( ) succss If D < 0 th th syst tds to bco lss bckloggd ood! If D > 0 th th syst tds to bco o bckloggd Bd! Th pcdg lyss s bsd o th o buffg ssupto (6). If std th ft od ssupto (6b) s ppld wht hpps? ( ) λ Avl t (λ) s hozotl l s Fg. 4.2 Oly o stbl pot t so ( ) < Oc byod th udsd ulbu pot ub of bckloggd pckts tds to cs wthout boud Howv fo λ << d odt th syst wll typclly opt th dsd stbl pot fo log pods of t 4

5 pd by J. Succ ECE 543 Octob Stblzd Slottd Aloh Ift od ssupto (6b) (0) fdbck If stt of bcklog s pfct ( thoughput s xzd sudo Bys Algoth ˆ All pckts cosdd bckloggd upo vl ( ) ) d () th succss ( ) ( ˆ ) { ˆ } Wt ( ˆ ) ˆ ( ˆ ) x ˆ k { λ ˆ k λ }. λ ( 2). ˆk fo fo dl o collso succss (4.7) Stbl fo λ < Nd to kow o stt λ Stblty du to ccut stt of : ˆ 0 : Wht th x-xs d y-xs lbls of Fg. 4.5? 5

6 pd by J. Succ ECE 543 Octob Appoxt Dly Alyss (sudo Bys) ˆ : succss fo 2 fo v ( ) succss W Totl dly fo th t of th th vl to th t of th th succssful tssso W R j t j y (4.8) Nub of bckloggd pckts t th t of th th vl R Rsdul dly of th cut slot t th t of th vl t j Dly du to j th pckt hd of th pckt y Rg tvl utl th bgg of th th xsso W Expctd uug dly E [ W ] Cobg E[ t j ] Lw ( λ W ): succss [ ] 2 [ y] E d Lttl s W 2 λ W E (4.9) 2 E[ y] W λ If ˆ th [ y] 0 R E d f ˆ 2 th [ y] E : 6

7 pd by J. Succ ECE 543 Octob E [ y] ( ) ( p ) ( ) ( λ p ) λ (4.0) λ p obblty of sgl bckloggd pckt I wost cs p 0 : E [ y] < Thus upp boud (ot gv txt) fo W s s follows: W 2 < λ By Expotl Bckoff sudo Bys ppoch ssus (0) fdbck Wht f od oly kows whth ts ow tsssos succssful? BEB soluto: Aft th usuccssful tssso: 2 E.g. Etht 7

8 pd by J. Succ ECE 543 Octob Uslottd Aloh Lk slottd Aloh cs xcpt tht y oth tssso th tvl fo t to t wll cus collso Assu do tssso tvl tht s xpotl wth pt x (x thfo s lso osso tssso ttpt t) ( ) x λ Cosdg od tht bgs tsttg t t t: A collso y occu f oth bg tsttg th pvous tssso tvl o If oth bgs tsttg th subsut tssso tvl Thus t s ud tht oly sgl tssso ttpt occus th tvl fo t to t ( ( ) ) obblty tht sgl tssso ttpt occus th tvl fo t to t ( ) obblty o tssso ttpt occus th tvl fo t to t 2 ( ) ( ) succss (4.3) Uslottd Aloh hs xu thoughput of ( 2) ( ) 2 wh 8

9 pd by J. Succ ECE 543 Octob Splttg Algoths (4.3) C slottd Aloh thoughput xcd? Assu pfct dt (0) fdbck d ft ods Suppos two pckts colld d upo dtctg ll oth ods pus: Ech colldg pckt s tsttd dpdtly wth pobblty 2 th wth pobblty 2 o of th pckts wll b tsttd succssfully th xt slot d th oth pckt y tsttd th subsut slot Wth pobblty 2 dl slot o oth collso occus whch cs wth pobblty 2 succssful tssso occu th subsut slot d so o Thus pobblty of 2 slots 2 pobblty of 3 slots 4 pobblty of slots ( ) 2 : 2 ( ) E[# slots fo tssso] E [ R] 2 [ R] ( ) p ( p) E p 2 E [ R] p ( p) p ( p) p p E p p p p [ R] ( p) 3 Two pckts 3 slots Thoughput of 3 2 ov th duto of tssso pod 2 9

10 pd by J. Succ ECE 543 Octob T Algoths Upo collso: Nods ot volvd t wt od Ivolvd ods doly slct th subst L tht tsts th xt slot o subst R tht wts A subsut collso would cu sl splttg of L to subst LL d subst LR so o E.g. Fg. 4.9: Slot Xt St Wtg St Fdbck S E 2 L R E 3 LL LRR 4 LR R E 5 LRL LRRR 0 6 LRR R E 7 LRRL LRRRR 8 LRRR R 9 R 0 How y pckts w volvd th collso slot? How y pckts blogd to st L d to st R? How y pckts blogd to st LL d to st LR? How y pckts blogd to st LRL d to st LRR? How y pckts blogd to st LRRL d to st LRRR? How c th lgoth b povd? 0