Problem Set 3 Solutions

Transcription

1 CSE Dsign n Anlysis of Switing Systms Jontn Turnr Prolm St Solutions. Consir t sign of port swit, s on suivi us wit knokout onntrtors t t output. Assum tt t links oprt t G/s n tt t iruit tnology n support mximum lok rt of MHz. Tis mns tt IPP must trnsmit lls on t lst tn prlll signl lins n tt OPP must riv lls on t lst input lins. T tn signl lins lving IPP n us in on of two wys. Eitr, n IPP n sn on ll t tim in prlll form ovr t tn signl lins, or n IPP n sn tn iffrnt lls onurrntly ovr t tn signl lins. In t first s, knokout onntrtor s inputs, wi r tn its wi. In t son s, knokout onntrtor s inputs, wi r on it wi. How mny outputs must t onntrtors v in of t two ss, to prou pkt loss proility of no mor tn wn t input lo is %? In t first s, t proility tt n output rivs i lls in givn yl is just n i n i n n i ( / ) ( / ), wr n=. T proility tt n rriving ll is isr is n n i n i = + ( i k) (/ n) ( / n) if t knokout onntrtor s k output, so w n to fin t i k i smllst vlu of k for wi n n i n ( ) (/ ) ( / ) i i k n n. T smllst su k is i= k + i wn n=. n In t son s, t proility of riving i lls is (/ n) ( / n) i t smllst vlu of k for wi n n i n ( ) (/ ) ( / ) i i k n n i= k + i k is wn n=. i n i, so w n to fin. T smllst su Wt os tis imply out t mmory nwit rquir t t output ports? Compr t ost of implmnting t knokout onntrtors in of t two ss. Wi sign o you tink is t ttr oi? Wy? T son sign rquirs totl of onntrtor outputs, so t nwit ntring t mmory is. G/s n t totl mmory nwit is. G/s. T first sign rquirs totl mmory nwit of G/s, lmost tims s mu. T tpts of t two signs v out t sm omplxity, wit t son ing prps littl lss. Tr is mor ontrol logi n for t son sign, ut t ovrll iffrn twn t two is likly to firly most. T son

2 sign is lrly t ttr oi. T rution in mmory nwit is vry signifint vntg.. Consir uss ll swit wit suivi us wit inputs n outputs. Suppos tt ll tim, ll rrivs on input wit proility p n tt ll is ssign rnomly to iffrnt output. Giv n xprssion for t proility tn givn ll tim, output rivs xtly on ll. ( p /) ( p /) = p( p /) Suppos t OPP t output s knokout onntrtor tt n forwr up to six lls pr ll tim into t OPP s ll uffr. Giv n xprssion proility tn givn ll tim, mor lls rriv t output tn n pl in t uffr. i i ( p /) ( p /) i= i Giv n xprssion for t proility tt n rriving ll is isr. ( / p) i= i ( i ) ( p /) ( p /) i. Consir rings ATM swit wit xtrnl links oprting t G/s. Assum tt t ring is its wi n tt t iruit tnology ing us supports pointtopoint lins oprting t up to MHz. If t swit s svn yts of intrnl r informtion to ll for sning it on t ring, ws t mximum vrg trffi lo tt n support on t outgoing links? Witout ovr, it woul./( ) = %, so wit ovr, is.*/(+)=./) or out.%. Assum t ring uss slott ring protool wit usy/il it, wr ring intrf wit ll to sn uss t first mpty slot tt ss to sn its ll. Suppos trffi is rriving t G/s t of inputs troug n tis rriving trffi is ll going to outputs troug. Wt frtion of its trffi is input tully l to sn? T ring n nl % of t trffi from links, so % of t trffi from. links. So inputs troug will l to forwr ll of tir input trffi, input will l to sn % of its input trffi n inputs n will unl to sn ny of tir trffi.. Consir sr uffr swit wit n inputs n n outputs. Assum tt tr is no limit on t vill uffr sp, tt t pkt lngts r xponntilly istriut, wit mn /µ n tt t tim twn pkt rrivls for outpus xponntilly istriut, wit mn /λ. Giv n xprssion for t proility tt t sr uffr ontins i pkts for prtiulr output. Unr t givn onitions, output n mol y n M/M/ quu, so t proility tt t sr uffr ontins i pkts for givn outpus ( ρ)ρ i wr ρ=λ/µ. i

3 Giv n xprssion for t proility tt t sr uffr ontins totl of i pkts, ssuming tt t numr of pkts going to t iffrnt outputs r inpnnt. In tis s, w v to onsir ll possil wys tt w n g pkts. So, t rquir xprssion is n i n i ( ρ ) ρ = ( ρ) ρ i + i + L+ in = i i + i + L+ in = i wr t summtion is ovr ll nonngtiv intgr vlus for i,...,i n tt up to i. If w ll tis quntity f(i,n), w n o t omputtion itrtivly using t qution wr f(j,)=( ρ)ρ j. f ( i, n) = j i f ( j,) f ( i j, n ) Us tis xprssion to stimt t mount of mmory n to nsur tt t pkt loss proility is no mor tn, wn n= n t input lo is %. Compr tis to t mount of mmory you woul n if t mmory wr not sr. A Visul Bsi progrm to omput f(,n) + + f(i,n) is sown low. Using tis, w fin tt t proility tt t quu ontins mor tn pkts is just unr. For port sr uffr swit, t proility tt t sr uffr s mor tn pkts is just unr. So t sr uffr swit ns storg for pkts, wil swit wit sprt uffrs for output woul n =, wi is out. tims s mny s t sr uffr swit ns. Not tt tis mto for omputing t rquir uffr siz ovrstimts wt s rlly n y smll mount. Funtion suf(ro As Doul, i As Intgr, n As Intgr) As Doul ' Comput t proility tt sr uffr swit wit n ports ' n uniform rnom input lo of ro, s lss tn or qul to ' i pkts in its uffr, ssuming no limit on t numr of pkts ' tt n stor. Dim, j, r As Intgr Dim pi(, ) As Doul Dim s As Doul pi(, ) = ro For j = To i pi(j, ) = ro * pi(j, ) Nxt j For = To n For j = To i pi(j, ) = For r = To j pi(j, ) = pi(j, ) + pi(r, ) * pi(j r, ) Nxt r Nxt j Nxt s = For j = To i s = s + pi(j, n) Nxt j suf = s En Funtion

4 Consir wt ppns wn TCP trffi is ppli to t sr uffr swit. Spifilly, ssum tt n/ of t outputs r riving trffi from lrg numr of TCP strms, wit lrg klog of trffi. Also ssum tt tr is singl quu in t sr uffr for of t outputs, n tt rriving pkts r isr if t quu is full. Qulittivly sri t quuing vior of t sr uffr swit in tis sitution. Estimt t mount of mmory n y t swit to nsur tt non of t usy output links xprins unrflow, ssuming tt t ntwork roun trip tim is ms n tt t output links oprt t G/s. Compr tis to t mount of mmory n in swit tt os not sr t mmory mong t iffrnt outputs. In tis sitution, t iffrnt TCP flows going out on ll t outputs n potntilly om synroniz, sin wn t sr mmory fills up, ll t flows r likly to xprin pkt loss n ru tir winow sizs togtr. To prvnt t sr uffr from unrflowing, w n mmory siz ts omprl to t prout of t output link nwit tims t ntwork rountrip ly. If t roun trip tim is ms n t output links r G/s, tis oms to out Mits for ongst link. If w llow for lfson s wort of uffring, tis numr grows to Mits pr ongst link. Sin lf of t links r ongst, tis rus t mmory usg y just ftor of, n sin mor tn n/ links oul ongst, it s not lr tt w woul vn gt tis rution, in prti. Consquntly for ontinuously klogg TCP flows, tr is littl gin otin from t sr uffr. Bs on ll your nswrs ov, vlut t vntgs n isvntgs of sr uffr swits, rltiv to swits wit sprt mmoris for output. Sr uffring works wll, so long s t trffi going to iffrnt outputs is unorrlt. Sin most TCP flows omplt tir t trnsmission for going troug multipl rouns of ongstion voin, tir vior is mor lik tt prit y t first nlysis, tn y t son. Tis suggsts tt tr is signifint rution in mmory tt n otin using sr uffring. Tis my not lwys l to signifint rution in ost, sin ommril mmory omponnts oftn o not v t il omintion of IO nwit n mmory pity. For ig sp routrs, mmory nwit is oftn t limiting ftor, foring routr signr to us mor mmory omponnts in prlll to gt t rquir nwit, ling to n ovrsupply of mmory pity. In tis sitution, tr is lss to gin from sring t mmory mong iffrnt ports.. Consir rossrs swit wit ports in wi IPP s singl quu n uring ritrtion yl, t IPPs ontn for t output tt t first ll in t quu is rss to. If vry input s ll in its quu n if t outputs ts lls r rss to r slt t rnom, ws t proility tt no lls r irt to prtiulr output? ( / n ) n = ( / ) =. Ws t xpt numr of outputs tt no lls r rss to?.n =. Suppos tt IPP s two quus, on for lls rss to vnnumr outputs n on for lls rss to onumr outputs. Assum tt vry IPP s ll in

5 of its two quus n tt t rsss of lls in t vn quus r rnomly slt from mong t vn outputs n tt t rsss of lls in t o quus r rnomly slt from mong t o outputs. Ws t proility tt givn output s no lls rss to it? ( / ) =. Ws t xpt numr of outputs tt no lls r rss to?.n =. Consir systm in wi IPP s su n ovn quu rrngmnt, n uring ritrtion yl, IPP ttmpts to sn ll from itr of its quus. Estimt t mximum trougput possil in su systm. In singl ll yl, w woul xpt.=. lls to gt troug, unr t givn onitions, ompr to out for t systm in wi IPP s singl quu. In susqunt ll yls, t rsss of lls will not inpnnt, ling to grtion in t trougput, ut t grtion soul lss in t s of n ovn quu systm tn in t s of singl quu systm. T tul mximum trougput soul twn % n % of, or rougly to lls pr yl.

6 . T timslott ritrtion ring, sri on pg, is ttrtiv us is vry simpl to implmnt. It n gnrliz to pply to systms wit VOQs n to llow suling ovr multipl tim stps. In tis gnrliz vrsion, t it z i is rpl y vlu, wi rprsnts t rlist tim stp t wi t output x i is not yt to riv ll. If inpu s n un ll to sn to output x i n s no otr ll to snt t tim, it s t witing ll for trnsmission t tim, n tn inrmnts for pssing its vlu to inpu. T input kps trk of wn witing ll is for trnsmission, n sns it t t pproprit tim. T mtrix sown low rprsnts st of lls witing to troug rossr swit using tis gnrliz timslott ritrtion ring. T ntry in row i, olumn j is t numr of lls npu going to output j. For input, list t output tt sns ll to on sussiv tim stp, until ll t lls r snt. Assum tnitilly, ll t vlus r zro n tt x i =i. T figur low sows wt ppns uring tim. In t firstrtion of t suling lgoritm, noting ppns, sin tr r zros in ll t igonl ntris. T first mtrix in t top row igligts wi VOQs r unr onsirtion uring stp of t lgoritm. T vlu of ftr stp r sown nxt, long wit t prtil onstrut up to tt point. T ntr mtrix in t top row is sows t numr of un pkts in of t VOQs n igligts t VOQs tt r unr onsirtion in t nxt stp. In t prtil s, t unrsor rtr (_) is us to init tim stps uring wi prtiulr inpus not to sn ll. t=,,,,, _,,_,,,,,,,, _,,_,,_,,,,,,_,,,,,, _,,_,,_,,,,

7 T figur low sows wt ppns t tim. Noti tt t initil rmovs tos lls tt wr to go out uring stp. Also not tt ti vlus tt r qul to t t n of tim r inrs to. Tis is nssry to nsur t r mks progrss. t=,_,,_,,,,,,,,,,_,,_, _,, _,,,,,_,,,,,,,,_, _,,,,_,_,,_,,,,,,,,_,,_, _,,,_,_,,,_,_,,,_,,,,,,,,_,,_,,_,_,_, _,,,_,_,,,,_,_,,,_,,,,,,,,_,,_,,_,_,_,,,,_,_,,,,_,_,, Tis figur sows wt ppns t t=. t= _,,,,,,_, _,,_,_,_,,,_,_,,,_,_,, _,,,,,,_,_,_,,_, _,,_,_,_,,,_,_,,,_,_,, _,,,,_,_,_,,,,_,_,_,,_, _,,_,_,_,,,_,_,,,_,_,,,_,,,,,_,_,_,,,,_,_,_,,_,, _,,_,_,_,,,_,_,,,_,_,,,_,,,,,_,_,_,,,,_,_,_,,_,, _,,_,_,_,,_,,,_,_,,,,_,,,_,,,,,,_,_,,,,_,_,_,,_,, _,,_,_,_,,_,,,,_,,,,_,,,_,

8 T figur low sows wt ppns in t nxt fw stps. Svrl mor tim stps r rquir to fully ll t lls. t=,,,,_,_,,,_,_,_, _,,,_,_,_,,_,,,_,,,_,,,_,,,,,_,_,,,_,_,_, _,,,_,_,_,,_,,,_,,,_,,,_,,,,,_,_,,,,_,_,,,,_,_,_,,_,,,_,,,_,,,_,,_,,,,,_,_,,,,_,_,,,,_,_,_,,_,,,_,,,_,,,_,,_,,,,,_,_,,,,_,_,,,,_,_,_,,_,,_,,,_,,,_,,,_,,_,,,,,,_,,,,_,_,,,,_,_,_,,_,,_,,,_,,,,,,_,,_, t=,,,,_,,,_,_,, _,_,_,,_,,_,,_,,,,,,_,,_,,,,,_,,,_,_,, _,_,_,,_,,_,,_,,,,,,_,,_,,,,,_,,,_,_,, _,_,_,,_,,_,,_,,,,,,_,,_,,,,,_,,,_,_,, _,_,_,,_,,_,,_,,,,,,_,,_,,,,,_,,,_,_,, _,_,_,,_,,_,,_,,,,,,_,,_,,,,,_,,,_,_,, _,_,_,,_,,_,,_,,,,,,_,,_, t=,,,_,,_,_, _,_,,_,,_, _,,,,,_,,_,,,,_,,_,_, _,_,,_,,_, _,,,,,_,,_,,,,_,,_,_, _,_,,_,,_, _,,,,,_,,_,,,,_,,_,_, _,_,,_,,_, _,,,,,_,,_,,,,_,,_,_, _,_,,_,,_, _,,,,,_,,_,,,,_,,_,_, _,_,,_,,_, _,,,,,_,,_,

9 . T figur low sows t stt of simpl rossr r t t strt of suling oprtion. Sow t stt of t ontrollr ftr stp of t lgoritm, until ll possil mts v n m. B sur to sow ow t pointrs r upt. Do tis for ot t roun roin lgoritm n t islip lgoritm. T rsult for t roun roin lgoritm is sown low. first roun outputs inputs slnputs slt outputs son roun outputs slnputs inputs slt outputs

10 T rsult for t islip lgoritm is sown low. first roun outputs inputs slnputs slt outputs son roun outputs slnputs inputs slt outputs

11 . Giv n xmpl sowing tslip n rquir up to itrtions to omplt pkt suling oprtion for port rossr. Spifilly, giv n initil stt of t rossr, sowing wi inputs v pkts to sn to wi outputs, n initil vlus of ll t pointrs us y t islip lgoritm. Tn sow wt ppns in of t rmining stps. In prtiulr, sow t stt of t pointrs n wi inputs v n mt to wi outputs, wn t lgoritm trmints. initil onfigurtion first stp son stp tir stp fourt stp fift stp

12 . T figur low sows t stt of rossr wit virtul output quus. In prtiulr, t numr in row i, olumn j nots t numr of lls witing in t VOQ npu going to output j. T numrs in t ottom row rprsnt t numr of lls in t quus t of t outputs. Sow ow t LOOFA lgoritm mts inputs to outputs, ssuming tt t outputs slnputs, s on wi input s t longst VOQ for t output. Us t xtr opis of t figur to sow wt mts r m ftr stp y irling t slt ntris. Mk sur you ll t igrms you r using to init stp. At t n, sow ow t sttus of t VOQs ngs s rsult of t slt ll trnsfrs. first stp son stp tir stp finl stt

13 . T figur low (similr to t figur on pg of t ltur nots) sows t initil stt of CCF rossr r t t strt of suling oprtion. Sow t stt of t ontrollr ftr stp of t lgoritm. nw rrivls following insrtion of rriving lls ftr n stp ftr t stp initil stt ftr st stp ftr r stp ftr t stp T stps r illustrt ov. Following t fift stp, t lgoritm trmints us stl mting s n foun.

14 . In on vrint of t LOOFA rossr suling lgoritm, outputs tt riv multipl is from inputs, slnputs s on t timstmps of t ontning lls (lls wit smll timstmps r prfrr ovr lls wit lrgr timstmps). Also, lls r forwr from t outputsi quus in timstmp orr. Fin trffi pttrn tt monstrts tt vn wit spup of, tis vrsion of t LOOFA suling lgoritm os not lwys forwr lls in FIFO orr. T figur low illustrts snrio in wi t olst lls first vrsion of t LOOFA lgoritm fils to prsrv FIFO orring. Tis is for six port swit. following rrivl stp stintion & timstmp stp,,, following trnsfr,,, following prtur,, following trnsfr timstmp of quu lls,, stp,,,,,,,,,,,,,,,,,,, stp,,,,,,,,,,,,,,,,, stp,,,,,,,,,,,, stp,,,,, Now, output must sn ll wit timstmp, wil input still s ll wit timstmp.

15 . Sow tt t stl mting lgoritm sri on pg os in ft prou stl mting. Assum, to t ontrry, tt t onstrut mting is not stl. Tis mns tt tr r two pirs (, ) n (, ) su tt prfrs to n prfrs to. If tis wr t s, tn must v m i for for it i for. Sin t mmrs of B only ng prtnrs to improv tir prfrn, t t tim tt i for, itr ws unmt or it ws mt wit prtnr tt rnk no igr tn. In tis s, must v swit prtnrs, mting it wit. But tis yils ontrition, sin woul not v swit from to, ftr oming mt wit.. On wy to implmnt multisn rossr swit wit VOQs is for t IPPs to opy rriving multist ll to t VOQs for ll outputs tt r to riv opis. Wit tis ppro, t multist lls ppr no iffrnt tn unist lls to t rossr. Sow tt rossr n forwr multist lls wit fnout of F in workonsrving fsion, using t LOOFA lgoritm n spup of F+. W strt y ssuming tt t swit oprts in yls of F+ pss. T first ps of yl is t rrivl ps, uring wi n rriving ll my pl in up to F iffrnt VOQs. T nxt F pss r trnsfr pss, uring wi lls r trnsfrr from inputs to outputs troug t rossr. T nxt ps is t prtur ps, uring wi lls r trnsmitt on t output links. T lst ps is n itionl trnsfr ps. W fin t slknss of ll just s on pg of t nots. W not tt t lmm on pg gnrlizs t multist s. As wit t unist s, trnsfr yl, t minimum slknss t n inpunrss y t lst on (unlss it s no lls). During t rrivl ps, ll s slknss n rs y t most F n uring t prtur ps its slknss n rs y t most. Sin tr r F+ trnsfr pss, tr n no nt rs in ll s slknss uring n F+ ps yl. W n lso giv vrsion of lmm on. In prtiulr, w n sow tt following t rrivl ps of ny stp, t slknss of ll is t lst (F). For lls tt rriv uring t first tim stp, tis is lrly tru, sin t outputs v no lls t tis point, n t most F lls gt insrt into VOQs uring t first stp. So, t ll wi is lsn t orring s slknss of (F). As on, w pro y inution. Lt ll tt ws pl in VOQ uring t rrivl stp of ps t. If tr r no lls tt pr n wr prsnt t t input for stp t, tn t rsult lrly ols, sin only t lls tt rriv uring stp t oul pr n tr r t most F of ts, xluing. So, suppos is ll tt ws prsnt for stp t n tt prs. Assum lso, tt ll otr lls prsnt for stp t tt pr lso pr. By t inution ypotsis, following t rrivl ps of stp t, t slknss of ws t lst (F). So, just for t rrivl ps of stp t, t slknss of must t lst. Now, onsir t lls pl in VOQs uring stp t. L t numr of ts lls tt pr. Tis mns tt ftr t rrivl ps of stp t, t slknss of is t ls. Sin tr t most (F)i otr lls tt rriv uring stp t tt pr n o not pr, t slknss of is t lst (i)((f)i)=(f). To omplt t proof tt t systm is workonsrving, suppos tt ftr t rrivl ps of som stp, tr is som output wit no lls in its output quu n tr r lls witing in VOQs for tt output. Lt ny su ll. Sin tr r no lls in t output quu, n sin t slknss of is t lst (F), tr n t most F lls tt pr ts input. Tis mns tt uring of t F trnsfr pss twn t rrivl ps n t prtur ps,

16 itr som ll in front of gts trnsfrr, or t input ttmpts to sn. Sin tr only F lls in front of, tr must t lst on trnsfr ps uring wi t input ttmpts to sn. If os not gt snt uring su trnsfr ps, tn t output must v gottn ll from som otr input. So, t output s ll to sn, uring t prtur ps.