Wide-Sense Nonblocking Clos Networks under Packing Strategy

Transcription

1 Wide-Sese Noblockig Clos Netwoks ude Packig Stategy Yuayua Yag Depatmet of Compute Sciece ad Electical Egieeig Uivesity of Vemot Buligto, VT 00 Jiachao Wag GTE Laboatoies 0 Sylva Road Waltham, MA 0 jwag@gtecom Abstact I this pape, we study wide-sese oblockig coditios ude packig stategy fo the thee-stage Clos etwok, o v(m; ; ) etwok Wide-sese oblockig etwoks ae geeally believed to have lowe etwok cost tha stictly oblockig etwoks Howeve, the aalysis fo the wide-sese oblockig coditios is usually moe difficult Mooe[] poved that a v(m; ; ) etwok is oblockigude packig stategy if the umbe of middle stage switches m This esult has bee widely cited i the liteatue, ad is eve cosideed as the wide-sese oblockig coditio ude packig stategy fo the geeal v(m; ; ) etwoks i some papes, such as [] I fact, it is still ot kow that holds fo v(m; ; ) etwoks whethe the coditio m whe I this pape, we itoduce a systematic appoach to the aalysis of wide-sese oblockig coditios ude packig stategy fo geeal v(m; ; ) etwoks with ay values We fist taslate the poblem of fidig the ecessay ad sufficiet oblockig coditios fo v(m; ; ) etwoks to a set of liea pogammig poblems We the solve this special type of liea pogammig poblems ad obtai a elegat closed fom optimum solutio We pove that the ecessay ad sufficiet coditio fo a v(m; ; ) etwok to be oblockig ude packig stategy is m j, F, k whee F, is the Fiboacci umbe We believe that the systematic appoach developed i this pape ca be used fo aalyzig othe wide-sese oblockig cotol stategies as well Itoductio The well-kow Clos etwoks[, ] have bee widely used fo multipocesso itecoectios ad data commuicatios Some ecet applicatios iclude the NEC ATOM switch desiged fo BISDN[], the IBM GF multipocesso[] ad ANSI Fibe Chael Stadad fo itecoectio of pocessos to the I/O system I geeal, a thee-stage Clos etwok with N iput pots ad N output pots has switch modules of size m i the iput stage, m switch modules of size i the middle stage,ad switch modules of size m i the output stage The etwok has exactly oe lik betwee evey two switch modules i its cosecutive stages Suppoted by the US Natioal Sciece Foudatio ude Gat No OSR-900 ad MIP-9, ad by the US Amy Reseach Office ude Gat No DAAH Such a thee-stage etwok is deoted as a v(m; ; ) etwok A geeal schematic of a v(m; ; ) etwok is show i Figue Output Stage Iput Stage Middle Stage m Figue : A geeal schematic of a v(m; ; ) etwok The coectig capability of the etwok i Figue is detemied by the paametes,, adm Foagive ad, vayig m, the umbe of middle switches, vaies its coectig capability Theefoe, the mai focus of the study o Clos etwoks has bee o fidigthe miimum value of the etwok paametem fo a cetai type of coectig capability to achieve the miimum etwok cost The v(m; ; ) etwoks have bee extesively studied i the liteatue We summaize some fudametal esults below Clos showed[] that a v(m; ; ) etwok is stictly oblockig if the umbe of middle switches m, If the etwok satisfies this coditio, ay middle switch may be chose abitaily to make a coectio without ay eaagemet of existig coectios i the etwok Fewe middle switches ae sufficiet if oly eaageability is eeded Bees showed[] that a v(m; ; ) etwok is eaageable if the umbe of middle switches m Ude this coditio, satisfyig a coectio equest may equie eaagemetof existig coectio i the etwok Sice the eaagemet causes disuptio of o-goig commuicatios ad time delay i path outig, oblockigcapability is geeally desiable Wide-sese oblockig capability ca be cosideed as a compomise betwee stictly oblockig capability ad eaageability Fo a wide-sese oblockig etwok,a itelliget outig cotol stategy must be employed to gove the pocess of path outig Though caefully selectig the paths used to satisfy the cuet coectio equest, the oblockig capability fo futue coectio equests ca be maitaied, ad at the same time lowe etwok cost ca be achieved Howeve, i geeal, the aalysis of wide-seseoblockig coditios is much moe difficult tha that of stictly oblockig coditios A commoly used outig cotol stategy fo wide-sese oblockig v(m; ; ) etwoks is the so-called packig stategy[] Ude packig stategy, a coectio is ealized o a path foud by

2 tyig the most used pat of the etwok fist ad the least used pat last Fo a v(m; ; ) etwok, this meas that whe choosig a middle switch fo satisfyig a coectio equest, a empty middle switch is ot used uless thee is ot ay patially filled middle switch that ca satisfy this coectio equest Packig stategy ca also be combied with epackig Repackig meas whe a coectio is eleased oe o moe existig coectios ae moved to the most used pat of the etwok Repackig essetially is a type of eaagemet although the eaagemet is pefomed at the time a coectio is eleased ot at the time a coectio is established It is geeally believed that packig/epackig ca impove etwok pefomace ad educe etwok cost Thee has bee some wok o packig/epackig outig i v(m; ; ) etwoks i the liteatue[8, 9, 0,, ] Ackoyd[8] ad Giad ad Hutubise[9] showed by simulatio that packig/epackig stategies have lowe blockig pobability ove adom outig Mu, Tag ad Devaaja[0] poposed a pobabilistic model fo the aalysis of blockig pobability ude packig stategy ad thei model cofimed the Ackoyd s simulatio esults Jajszczyk ad Jekel[] deived the ecessay ad sufficiet coditio fo a epackable v(m; ; ) etwok ad showed that a epackable etwok equies fewe middle switches tha a stictly oblockig etwok Howeve, the theoetical aalysis of the oblockig coditios ude packig stategy is difficult The oly kow esult coceig packig stategy is Mooe s wok (cited i Bees book[]) Mooe[] aalyzed the wide-sese oblockig coditios ude packig stategy fo v(m; ; ) etwoks, ad showed that a v(m; ; ) etwok is oblockig ude packig stategy if m, which is a sigificat impovemet ove the stictly oblockig coditio m, This esult has bee widely cited i the liteatue, eg [, ] I some papes, such as [], it is eve cosideed as the wide-sese oblockig coditio ude packig stategy fo the geeal v(m; ; ) etwoks with ay values I fact, it is still ot kow that whethe the coditio m holds fo v(m; ; ) etwoks whe, ad that if the aswe is egative, what the wide-sese oblockig coditios ude packig stategy ae fo the geeal v(m; ; ) etwoks Mooe s appoach[] is based o eumeatig all possible states of a middle switch ad cosideig all state tasitios As ca be see i the ext sectio, the umbe of the states of a middle switch gows vey fast as iceases Theefoe, it is difficult to diectly exted Mooe s appoach to aalyzig wide-sese oblockig coditios fo geeal v(m; ; ) etwoks fo I this pape, we will itoduce a systematic appoach to the aalysis of wide-sese oblockig coditios ude packig stategy fo geeal v(m; ; ) etwoks with ay values We will fist taslate the poblem of fidig the oblockig coditios fo v(m; ; ) etwoks to a set of liea pogammig poblems We will the show that fo this special type of liea pogammig poblems, we ca fid a elegat closed fom optimum solutio ad thus obtai the oblockig coditios fo the oigial poblem Coectio Requests ad Netwok States I this sectio, we povide some defiitios o coectio equests ad etwok states of a v(m; ; ) etwok Fo ay i; j f;;:::;g, we deote a coectio equest fom a iput of iput switch i to a output of output switch j i a v(m; ; ) etwok as (i; j) Based o the etwok stuctue i Figue, if coectio equest (i; j) is satisfied, it must go though a middle switch fom iput i of the middle switch to output j of the middle switch I this case, we say coectio (i; j) passes this middle switch Fo a middle switch, thee will be up to mutually disjoited coectios of the fom (i ;j );(i ;j );:::;(i k;j k),whee k ; i <i < <i k, j ;j ;:::;j k, ad fo ay s; t k, j s = j t whe s = t I Mooe s appoach[], the state of a middle switch was defied as all coectios passig that middle switch By a simple calculatio, P, we ca obtai that the umbe of diffeet such states is [, P i=0 i i i!] = i=0, i!foa i middle switch As gets lage, the umbe of states iceases damatically Thus, it is difficult to diectly geealize Mooe s appoach to geeal v(m; ; ) etwoks with ay values I the followig, we itoduce a systematic appoach to the aalysis of wide-sese oblockig coditio ude packig stategy fo geeal v(m; ; ) etwoks NONE (,) (,) (,) (,)(,) (,)(,) Figue : Seve possible middle switch states fo = Without loss of geeality, i the est of the pape, we will always assume that the cuet coectio equest to be satisfied is (; ), that is, we ae about to coect a iput of iput switch to a output of output switch We defie the middle switch states with espectto coectioequest (; ) as follows Defiitio Fo a middle switch, the switch is said to be i state [; ], if ad oly if thee exists a coectio (; ),othee exist two coectios (;j)ad (i; ) i the switch whee i; j = Defiitio Fo a middle switch, the switch is said to be i state [;j]whee j =, if ad oly if thee exists a coectio (;j)ad thee does ot exist ay coectio (i; ) fo ay i i this switch Defiitio Fo a middle switch, the switch is said to be i state [i; ] whee i =, if ad oly if thee exists a coectio (i; ) ad thee does ot exist ay coectio (;j)fo ay j i this switch Defiitio Each of the states defied i Defiitios is efeed to as a fobidde state with espectto coectio equest (; ), ad all othe states ae efeed to as o-fobidde states with espect to coectio equest (; ) Clealy, accodig to the above defiitios, a ew coectio equest (; ) caot be satisfied by a middle switch i a fobidde state Also, the above defiitio o the states of a middle switch is uique I othe wods, if a middle switch is said to be i state [i ;j ]ad i state [i ;j ], the it must be that i = i ad j = j Theefoe, fo a middle switch, thee ae a total of diffeet states, amog which, ae fobidde states I the followig, we will always use these defiitios o the states of a middle switch istead of the oe used i Mooe s appoach[] (,)

3 The Geeal Noblockig Coditio I this sectio, we deive the geeal oblockig coditio suitable to ay cotol stategies Sice we ae about to satisfy coectio equest (; ), weae iteested i those middle switches i states [; ], [; ], :::, [; ], [;], [;], :::,[;],ad [;] Fo ay i; j f;;:::;g, we deote the umbe of middle switches i state [i; j] as x i;j Sice coectio equest (; ) caot be satisfied by ay middle switch i states [i; ], [;j] (i; j = ), o [; ], it will be satisfied by a middle switch i a o-fobidde state Thus, the umbe of Pmiddle switches equied fo satisfyig coectio equest (; ) is P i= xi; + x;j + x; + : j= We have the followig theoem coceig the geeal oblockig coditio of a v(m; ; ) etwok Theoem The ecessayad sufficiet coditio fo a v(m; ; ) etwok to be oblockig is m max ( X i= x i; + X j= x ;j + x ; ) + ; whee, is a set of costaits fo vaiables x ;, x i; ad x ;j (i; j = ) Poof Sice the umbe of middle switches that caot be used fo satisfyig coectio equest (; ) P P is o moe tha i= xi; + x;j + x;, oe moe middle switch that is j= defiitely i a o-fobidde state ca be used fo ealizig coectio equest (; ) Theefoe, the sufficiecy holds O the othe had, let x i;, P x ;j (i; j P = ) ad x ; equal to some values that make ( i= xi; + x;j + x;) to achieve j= its maximum ude costaits The we eed at least max P i= xi; + P j= x;j + x; o + middle switches to satisfy coectio equest (; ) Thus, the ecessity holds Clealy, costaits ae cotol stategy depedet Diffeet costaits will lead to a diffeet type of oblockig capability I the followig, we itoduce some geeal costaits that ae suitable to ay cotol stategies fo the v(m; ; ) oblockig etwoks We have the followig lemma that is useful fo establishig the geeal costaits fo oblockig v(m; ; ) etwok Lemma I a v(m; ; ) etwok, at ay time, amog all existig coectios passig middle switches, thee ae o moe tha coectios of the fom (i; ) (ie statig fom some iput of iput switch i), ad thee ae o moe tha coectios of the fom (;j) (ie coected to some output of output switch j), fo ay i; j f;;:::;g Poof Notice the fact that thee ae iputs o iput switch i ad outputs o output switch j i a v(m; ; ) etwok By Lemma, we kow that befoe coectio equest (; ) is satisfied, thee ae at most, existig coectios of the fom (; ) ad at most, existig coectios of the fom (; ) We ca obtai two geeal costaits as follows X i= x i; + x ; + : () X j= x ;j + x ; + : () Note that ude ay cotol stategy, a wide-sese oblockig v(m; ; ) etwok should always satisfy geeal costaits () ad () I additio, ude diffeet cotol stategies, it should also satisfy some stategy-depedet costaits The Costaits ude Packig Stategy I this sectio, we deive additioal costaits fo vaiables x ;, x i; ad x ;j (i; j = ) whe packig stategy is employed; that is, we ivestigate the elatioship amog x i; s ad x ;j s ude packig stategy Note that the costaits deped o the histoy of the etwok states o the ode i which the coectio equests ae satisfied We defie a coectio sequece as follows Defiitio Let S be the set of, coectios (;), (; ), :::, (;),(;),(;),:::,(;) Fo the states of middle switches at a time, we defie a coectio sequecep p :::p, coespodig to the ode of states [i; ] ad [;j](i; j = ) as follows p p :::p, is a pemutatio of the, coectios i S; The coectios i S ae pemuted to p p :::p, i the ode of the last state tasitio of the coespodig state I othe wods, fo ay k ad k, p k = (i ;j ) pecedes p k = (i ;j ) i the coectio sequece p p :::p, if ad oly if the last middle switch i state [i ;j ] eached its cuet state befoe the last middle switch i state [i ;j ] eached its cuet state We ow i the positio to deive the costaits ude packig stategy Lemma Assume that thee ae middle switches cuetlyi states [i ; ]; [i ; ], :::;[i k;], i ;i ;:::;i k If, at this time, satisfyig a coectio equest (;j)(j =)though a middle switch will cause the state of the switch to chage to [;j],the afte this coectio is satisfied, we have the followig costait s= x i s; + x ;j : () Similaly, assume that thee ae middle switches cuetly i states [;j ];[;j ], :::;[;j k], j ;j ;:::;j k If, at this time, satisfyig a coectio equest (i; ) (i = )though a middle switch will cause the state of the switch to chage to [i; ], the afte this coectio is satisfied, we have the followig costait s= x ;j s + xi; : () Poof Assume that thee ae middle switches cuetly i states [i ; ], [i ; ], :::, [i k;] ad at this time satisfyig a coectio equest (;j) though a middle switch will cause the state of the switch to chage to [;j] This implies that coectio equest (;j)could ot be satisfied by ay middle switches i states [i ; ],

4 [i ; ], :::, [i k;] By packig stategy, sice coectio equest (;j)could ot be satisfied by a middle switch i state [i ; ],thee mustbe somecoectioof the fom (;j)i that switch Similaly, thee must be some coectio of the fom (;j)i the switches i states [i ; ];:::;[i k;] By Lemma, the umbe of coectios of the fom (;j)is o moe tha Thus, the costait () holds Similaly, we ca pove the secod pat of the Lemma Lemma Fo a coectio sequece p p :::p,, we have a total of, costaits fo, vaiables x ;, x i; ad x ;j (i; j = ) Poof Coside p t fo ay t Without loss of geeality, let p t = (;j) If p p :::p t, cotai (i ; ), (i ; ), :::, (i k;) whee k t,, by Lemma we have the costait s= x i s; + x ;j : Othewise, if p p :::p t, do ot cotai ay (i; ) fo i =, we have the costait x ;j : Symmetically, fo p t = (i; ) we ca obtai the coespodig costait Thus, each p t (t ) coespods to a costait By addig geeal costaits () ad () obtaied i the pevious sectio, we have a total of, costais fo, vaiables x ;, x i; ad x ;j (i; j = ) The poof of Lemma i fact tells us how to wite these costaits fo a coectio sequece as well The Necessay ad Sufficiet Coditio We have the followig impotat theoem coceig the oblockig coditios ude packig stategy fo a v(m; ; ) etwok Theoem The ecessayad sufficiet coditio fo a v(m; ; ) etwok to be oblockig ude packig stategy is m max f max pp Apxb c T xg; () whee, c, x ad b ae legth (, ) vectos with c =[ ::: ] T ; x = [x x ::: x,] T = [x ; x ; ::: x ; x ; x ; ::: x ; x ; + ] T ad b = [ ::: ] T, A p is a (, ) (, ) matix, A px b epesets the, costaits fo the coectio sequece p descibed i Lemma, P is the set of all coectio sequeces, that is, all possible pemutatios of, coectios (; ); (; );:::;(;);(;);(;);:::;(;) Poof Apply Lemma ad Lemma to Theoem Theoem taslates the poblem of fidig m that is ecessay ad sufficiet fo a v(m; ; ) etwok to be oblockig ude packig stategy to the poblem of fidig the maximum objective value amog a set of liea pogammig poblems I the ext two sectios, we will show how to fid the optimum solutio of the supe liea pogammig poblem () i two steps: Step Fo a give coectio sequece p i a v(m; ; ) etwok, fid the closed fom optimum solutio of the coespodig set of liea pogams max c T x: Apxb Step Fid the maximum value amog all the optimum values of the above liea pogammig poblems fo all possible coectio sequeces i a v(m; ; ) etwok Solvig the Liea Pogammig Poblem I this sectio, we fid the optimum solutio of the set of liea pogams fo a give coectio sequece Thee ae umeous liea pogammig books, eg [, ], discussig stadad methods of solvig liea pogammig poblems I geeal, fo a LP poblem thee may ot be a closed fom optimum solutio Howeve, as will be see late, fo this type of LP poblem, by employig some special techiques we ca obtai a elegat closed fom optimum solutio Due to the limited space, we omit o simplify the poofs of some lemmas ad theoems The detailed poofs ca be foud i a techical epot[] Pelimiaies Notice the followig lemma that ca simplify ou liea pogammig poblem Lemma The liea pogammig poblem max Apxb c T x; () whee A p;c, ad b =[ ::: ] T ae descibed i Theoem, ca be tasfomed to the liea pogammig poblem max Apxb c T x; () whee, A p ad c ae the same but b =[ ::: ] T Lemma allows us to simplify the costaits ()-() by eplacig o thei ight had sides by, ad o thei left had sides (if ay) by The followig lemma is useful i solvig ou special LP poblem Lemma If x 0 is a solutio to the system of liea equatios Ax = b ad y 0 is a solutio to the system of liea equatios y T A = c T,thex is a optimum solutio of the LP max c T x; (8) Axb y is a optimum solutio of the dual of the LP (8) ad c T x = y T b: mi b T y; y T Ac T y0 Poof Clealy, x is a feasible solutio of the LP (8) ad y is a feasible solutio of the dual LP O the othe had, we have that c T x = y T Ax = y T b: Theefoe, give ay feasible solutio x of the LP (8), we have c T x = yt A y x T b = c T x : That is, x is a optimum solutio of the LP (8) Similaly, y is a optimal solutio of the dual LP

5 A example fo = Befoe we solve the geeal fom of the LP (), let s coside a example fo = (a) Figue : Two coectio sequeces fo = (a) Coectio sequece p (b) Coectio sequece p 0 Fist, coside the coectio sequece p =(;)(;)(;) (;)(; )(; ) show i Figue (a) By Lemmas ad, we ca obtai the followig costaits (usig the simplified fom stated i Lemma ) (b) x ; + x ; x ; + x ; + x ; x ; + x ; + x ; + x ; x ; + x ; x ; + x ; + x ; x ; + x ; + x ; +(x ; +=) x ; + x ; + x ; +(x ; +=) x ;;x ;;x ;;x ;;x ;;x ;;x ; 0: Let x = x ;, x = x ;, x = x ;, x = x ;, x = x ;, x = x ;, adx = x ; + The costait matix i the LP () is A p = It ca be veified that the A px = b has a uique solutio x = 8 ;x = ;x = ;x = ;x = ;x = ;x = ; ad that the y T A p = c T has a uique solutio y = ;y = ;y = ;y = ;y = ;y = 8 ;y = : By Lemma P the optimum value of the LP () fo coectio sequece p is xi = i=, ad by Lemma the optimum value of the LP () fo coectio sequece p is Now coside aothe coectio sequece p 0 = (;)(;) (;)(;)(; )(; ) show i Figue (b) It is easy to check the costait matix i the LP () is A p 0 = It ca be veified that the A p 0x = b has a uique solutio x = 0 ;x = 0 ;x = 0 ;x = 0 ;x = 0 ;x = 0 ;x = 0 ; ad that the y T A p 0 = c T has a uique solutio y = 0 ;y = 0 ;y = 0 ;y = 0 ;y = 0 ;y = 0 ;y = 0 : By Lemma P the optimum value of the LP () fo coectio sequece p 0 9 is xi = i= 0, ad by Lemma the optimum value of the LP () fo coectio sequece p 0 is 9 0 By Theoem, to fid the oblockigcoditio fo a v(m; ; ) etwok, we eed to exam all possible pemutatios of coectios (; ), (; ), (; ), (; ), (; ) ad (; ) Howeve, fo this speciallp (), we may make use of the symmety amog coectio sequeces Fo example, it is clea that if we otate (; ), (; ) ad (; ) i p 0 to (; ), (; ) ad (; ) ad obtai a ew coectio sequece p 00 =(;)(; )(; )(; )(; )(; ), thep 00 has the same optimum value as that fo p 0 Afte cosideig all coectio sequeces, it tus out that is the maximum value amog all optimum values of the LP () Thus, the oblockig coditio i Theoem is m fo v(m; ; ) etwok The geeal fom of costait matix A p Now we detemie the geeal fom of costait matix A p i the LP () fo a coectio sequece p = (i ;j )(i ;j ):::(i,;j,) Based o the symmety of coectio sequeces ad fo pesetatioal coveiece, without loss of geeality, we assume that coectios (; ); (; );:::;(;) keep thei elative ode i p ad coectios (; ); (; );:::;(;) also keep thei elative ode i p, but (; ); (; );:::;(;)may be itelaced with (; ); (; );:::;(;) Also assume that (; ) is the fist coectio i the coectio sequece p, ie (i ;j ) = (;) The coectio sequeces p = (;)(; )(; )(; )(; )(; ) ad p 0 =(;)(; )(; )(; )(; )(; ) i Figue ae examples of such coectio sequeces Ude the above assumptios, we list all costaits fo coectio sequece p as follows Fo coectio (;j), j, if (;), (; ),:::, (k,;), (k ;)ae all (; ) s pecedig (;j) (we kow at least (; ) is ahead of (;j)), fom () ad Lemma, we have x ;j + k X x s; : (9) s= Fo coectio (i; ), i, if(; ), (; ), :::, (;k,), (;k ) ae all (; ) s pecedig (i; ), fom () ad Lemma, we have k X t= x ;t + x i; : (0)

6 Clealy, if o (; ) pecedes (i; ) i p, the costait will be educed to the tivial case x i; The last two costaits ae diectly fom (), () ad Lemma X X j= i= x ;j +(x ; +=) () x i; +(x ; +=) : () Thus, the (, ) (, ) costait matix A p ca be witte as " I H 0, # A p = L S e, : 0 T, T, I the above matix, 0,,, ad e, ae legth vectos with 0, = [00 ::: 0] T,, = [ ::: ] T ad e, = [0 0 ::: 0] T,I, S, L ad H ae (, ) (, ) matices, whee, I is a idetity matix, S = 0 0 ::: ::: ::: ::: 0 Lad H will be descibed below It is easy to see that fo a diffeet coectio sequecesp, A p diffes oly i L ad H We ow exam matices L ad H i moe detail Coside (;j )ad (;j )i the coectio sequece p,whee j j If(; ), (;),, (k ; ) ae all (; ) s pecedig (;j ) ad (; ), (; ),, (k ; ) ae all (; ) s pecedig (;j ),the we must have k k That is, H (as well as L by a simila agumet) is a staicase matix with s i the bottom-left pat ad 0 s i the up-ight pat as show i Figue 0 ; Figue : A staicase matix Pecisely, we have L = [L T T T L ::: L, ] T ; whee, L ;L ;:::;L, all ae legth (, ) ow vectos of the fom [ ::: 00 ::: 0]with l, l, :::, l, cosecutive s o the left espectively, ad 0 l l l, =, Note that l, =, is deived fom the costait () Similaly, H =[H H ::: H,];whee, H ;H ;:::;H, all ae legth (, ) colum vectos of the fom [0 0 ::: 0 ::: ] T with h, h, :::, h, cosecutive s at the bottom espectively, ad, = h h h, 0 Note that h =, is deived fom the assumptio that (; ) appeas fist i the coectio sequece p ad by the costait (9) Moeove, L ad H ae closely elated Coside (i; ) ad (;j) i the coectio sequece p,whee, i ad j Eithe (i; ) pecedes (;j)o (;j)pecedes (i; ) If(i; ) pecedes (;j),thex i; will appea i the costait (9) coespodig to this (;j), but x ;j will ot appeai the costait (0) coespodigtothis(i; ) O the othe had, if (;j)pecedes (i; ), the x ;j will appea i the costait (0) coespodig to this (i; ), but x i; will ot appea i the costait (9) coespodig to this (;j) Theefoe, matices L ad H have the followig foms L = H = t ; t ; ::: t ;, t ; t ; ::: t ;, t,; t,; ::: t,;, ::: ; t ; t ; ::: t,; t ; t ; ::: t,; t ;, t ;, ::: t,;, t ;, t ;, ::: t,;, whee, t i;j ad t i;j f0;g,adt i;j is the complemet of t i;j Also, by the above assumptios, we have t i;j t i,;j ad t i;j t i;j+: That is, we ca establish the followig elatios ad have L H = h i+ + l i =, ; i = ; ;:::;,; l 0 0 ::: 0 l l,l 0 ::: 0 l l,l l,l ::: 0 l, l,,l l,,l ::: l,,l, () Note that l ;l ;:::;l, ae iteges ad satisfy 0 l l l, =, : I fact, we ca show[] that the above coditio ca be educed to l l l, =,:I the est of the pape, we will always assume l, ad we efe to a coectio sequece with all l i asa o-tivial coectio sequece The solutio of the LP poblem ; The followig theoem gives the mai esult of this sectio Theoem Fo a o-tivial coectio sequece p, the system of liea equatios P A px = b has a uique solutio x, which satisfies x, 0 ad x i= i =, jdet(ap)j : Befoe we fomally pove Theoem, we eed to fist simplify the system A px = b We solve the system of liea equatios A px = b by Gaussia elimiatio[] ad elimiate x, fist Afte we elimiate x,, the system A px = b is equivalet to ad I L H Λ, x, = X X = X, i= x i = 0, X i= () x i ()

7 whee, X = x x x, Λ = 0 0 ::: ::: ::: ;,,, :::, ;X = x x + x, ; = ;0 = Lemma The system of liea equatios () is equivalet to ad 0 (L H, Λ)X =(L, 0); () It is staightfowad to veify that jdet(a p)j = det X =, H X : () I L H Λ ad by (), system () becomes = = jdet(l H, Λ)j (8) l, 0 ::: 0 l l,l, ::: 0 l, l,,l l,,l, l, 0 l, 0,l l0,,l ::: l0,,l, l, l, l,, l 0,, X (9) whee, l, 0 = l, + ad l l l, l, =,: The followig theoem gives a solutio to a system of liea equatios of moe geeal fom tha the system of liea equatios (9) Theoem Let A = X = which satisfy a, a a, a k,; a k,; ::: a k,;k,, ; a k; a k; ::: a k;k, a k;k x x x k, x k ; d = a, a, a k,;, a k;, : () a i;j, a i 0 ;j = a i;i 0+ fo j i 0 <ik, () f (i), the detemiat of the uppe left (i i) sub-matix of A (i paticula, f (k) =det(a)) is oe-zeo fo i k The the system of liea equatios AX = d has a uique solutio X ad T X = x i =, det(a) : i= Especially, if a ad a i;j 0 fo j i k, we have f (i) > 0 fo i k ad X 0 Poof (sketch) We simplify AX = d by Gaussia elimiatio ad tasfom A to a uppe tiagula matix Let ad Q j = g(i; j) = jx l= a i;lf (l, ); j i k;, g(j+;j) f(j), g(j+;j) f(j), g(k;j) f(j) whee j k, Multiplyig both sides of AX = d by Q ;Q ;;Q k, o the left yields AX e = d e with ea = ea, ea, ea k,;k,, ea k;k e ; d = ; ed d e ed k, d ek wheeea i;i = f(i,) f(i) fo i k, d e = a,, ad di e = a i;i f(i,) fo i k The we ca have a explicit expessio of A e,, ad obtai the solutio X = f() f() f() f() Fially, we ca show that f() f() f() f() f() T X = f (), + f () =, f () f(k) f() f(k) f() f(k) f(k,) f(k) f (i), f (i, ) a, a f() a f() a kk f(k,) f (i, )f (i) i= + f (i, ), f (i) i= =, f (k) =, det(a) : :

8 Poof of Theoem Note that the matix i (9) has the fom of A i Theoem Fo otatioal coveiece, we let l 0 = 0 ad deote l, 0 as l, Thus, we have a i;j = l i, l j, 0fo j i, ada = l, l 0 = l We fist show that (9) o its equivalet () satisfies coditio () i Theoem Sice a i;j = l i, l j, fo j i,, we have a i;j, a i 0 ;j =(l i,l j,),(l i 0,l j,)=l i,l i 0 =a i;i 0+ fo j i 0 <i, Thus, the coditio () i Theoem holds We pove coditio () by iductio o i Fist, wehave f() =a ; =l >0 Assume f (k) > 0foallk i, Note that a i;;a i;;:::;a i;i 0, ad at least oe of them is geate tha 0 (eg a i; = l i, l 0 = l i > 0) We ca show [] that f (i) =P i ai;jf(j, ): Thus, f (i) > 0 The coditio () i j= Theoem holds Applyig Theoem ad (8) to (9), we have that the system (9) o () has a uique solutio X 0 ad satisfiesp,,, X jdet(ap)j, X By () ad (), we have x i = x i =, jdet(a p)j i= i= Theefoe,, X i= x i =, jdet(a p)j + ad x, = i= x i = jdet(a p)j > 0: jdet(a =, p)j jdet(a : p)j The emaiig is to check X 0 I fact, sice all elemets i H ae 0 s o s, fom () we have fo ay i, i, P, x, i, j= x j = jdet(ap)j > 0: Similaly, we ca apply the same techique to the system of liea equatios y T A p = c T ad obtai the followig theoem Theoem Fo a o-tivial coectio sequece p, the system of liea equatios y T A p = c T has a uique solutio y,which satisfies y 0 Applyig Theoems ad to Lemma, ad by Lemma, we have the followig theoem givig the closed fom optimum values of LP () ad LP () Theoem The optimum values of the LP () ad the LP () ae, ad, ; espectively jdet(ap)j jdet(ap)j Now, let s apply Theoem to the coectio sequeces i the example of Sectio Fo coectio sequece p = (;)(; )(; )(; )(; )(; ), we ca easily veify that jdet(a p)j = Theefoe, the LP () fo p has the opti-, mum value, = Fo coectio sequece p0 = (; )(; )(; )(; )(; )(; ), we ca see that jdet(a p 0)j = 0 Thus, LP () fo p 0 has the optimum value,, 0 = 9 0 Maximum Objective Value amog a set of LP Poblems The maximum objective value amog the set of LPs that coespod to all possible coectio sequeces ca be witte as max f max pp Ap xb c T xg = max pp, jdet(a p)j =, : (0) max pp fjdet(a p)jg Thus, the poblem is ow tasfomed to the poblem of fidig the maximum value amog a set of detemiats Obseve the elatios i (8) ad (9), we eed to maximize the followig detemiat l, 0 ::: 0 l l,l, ::: 0 l, l,,l l,,l, l, l,,l l,,l ::: l,,l, () whee, l l l, <l, =: Now we simplify this detemiat by some ow/colum tasfomatios without chagig the value of the detemiat The followig tasfomatios ae pefomed i ode: subtact ow (,) fom ow (, ), subtact ow (, ) fom ow (, ), :::,ad subtact ow fom ow ; the subtact colum fom colum, subtact colum fom colum, :::, ad subtact colum (, ) fom colum (, ) Let = l +, = l, l +, = l, l +, :::,, = l,,l, +ad, = l,, l, + : Note that l P, l,, l,, l i, l i, 0fo i, ad, l + (li, li,)=letk =, The the detemiat i= () is equivalet to =,,,, k,,, k () whee i, i k, is a itege ot less tha, ad P k i= i = k, Theoem The optimum value fo P max k i= i=k, i Z+ whee, is the detemiat i () ad Z + is the set of all o-egative iteges, is achieved at ( ; ;:::; k,; k) = (;;:::; ; ), o symmetically, (; ;:::; ; ) We ae iteested i the value of the k k detemiat that achieves the optimum value i Theoem That is, we eed to evaluate f (k) =,,,,,, It is easy to see that f () =; ad f () =:Expadig o the last ow of this detemiat, we ca establish the followig ecuece f (k) =f(k,),f(k,); fo k : :

9 It is iteestig to ote the elatioship betwee f (k) ad the wellkow Fiboacci umbes F [] The Fiboacci umbes ae defied by the ecuece F 0 = 0; F = ; F = F, + F,; fo I fact, otice that F k+ = F k + F k, = F k, + F k, = F k, +(F k,,f k,)=f k,,f k,; ad F k+ = F = ifk =, F k+ = F = ifk = : Thus, we have f (k) =F k+; fo k : () Now, we ca obtai the followig closed fom oblockig coditio fo v(m; ; ) etwoks Theoem 8 The ecessayad sufficiet coditio fo a v(m; ; ) etwok to be oblockig ude packig stategy is m, F, whee, F, is the Fiboacci umbe Poof Fom (0), (), () ad (), ad otig that k =,, we have that the maximum objective value amog the set of LPs that coespod to all possible coectio sequeces max f max pp Apxb c T xg = ;, F, () whee, F, is the Fiboacciumbe Also, ote that x i;j s i x ae the umbes of middle switches i state [i; j] ad theefoe should be iteges, ad they should also satisfy costaits Ax b Thus, fom () ad Theoem, we obtai the oblockig coditio j k m, : F, Fom Theoem 8, we ca see that whe =, we have m, which agees with the wide-sese oblockig coditio 9 ;adwhe =, ; etc Also ote that whe F,, we obtai obtaied by Mooe[]; whe =, m m m,, which agees with the stictly oblockig coditio obtaied by Clos[] This idicates that, i tems of umbe of middle switches equied fo oblockig, packig stategy woks bette i the case of small ad has o advatage ove adom outig (stictly oblockig) whe F, 8 A Example fo the Necessay Coditio I this sectio, we demostate a example of coectio sequece that eaches the ecessay coditio i Theoem 8 Coside = Give a coectio sequece p =(;)(;) (;)(; )(; )(; ), we ca obtai all costaits ad matix A as show i Sectio The by usig the techiques discussed i Sectio, we ca solve the LP () ad obtai the optimum solutio x = [x ; x ; x ; x ; x ; x ; x ; + ] h T 8 i T = : Table : The sequece of coectio / discoectio opeatios fo the example Coectio/discoectio opeatios: Realize x ; coectios (; ) i Realize x ; coectios (; ) i Realize x ; coectios (; ) i Realize x ; coectios (; ) i Release the coectios i Step Realize x ; coectios (; ) i Release the coectios i Step 8 Realize x ; coectios (; ) Note that i this case, thee ae seveal choices ad we put them i 9 Realize x ; coectios (; ) Thee ae seveal choices ad we put them i 0 Realize x ; coectios (; ) i Release the coectios i Steps 8 ad 9 Realize x ; coectios (; ) i Realize x ; coectios (; ) i Realize x ; coectios (; ) i Release the coectios i Steps ad Realize x ; coectios (; ) i Realize x ; coectios (; ) i 8 Realize x ; coectios (; ) i 9 Realize x ; coectios (; ) i Goup (F) 0 Release the coectios i Steps, ad 8 Realize x ; + coectios (; ) i Goup (G) Recall that x i;j deotes the umbe of middle switches i state [i; j] I the followig, we show how to actually each this etwok state ude packig stategy, that is, how to each a etwok state with appopiate umbes of middle switches i the coespodig states We stat with a empty etwok, that is, o ay coectios i the etwok We the geeate a sequece of coectio/discoectio equests ad ealized them ude packig stategy as show i Table The coespodig middle switch states ae show i Figue I paticula, Figue (i) shows the middle switch states immediately afte Step i Fo pesetatioal coveiece, we ame the middle switches i the same state a goup Note that all steps i Table follow packig stategy ad afte P Step we have a total of xi;+p x;j +(x;+)= i= j= o-empty middle switches O the othe had, we ca see that fo = the Fiboacciumbe F, = F = ad the oblockig coditio is m 9 Coclusios, F, =,, = : I this pape, we have studied wide-sese oblockig Clos etwoks, o v(m; ; ) etwoks, ude packig stategy We itoduced a systematic appoach to the aalysis of wide-sese oblockig coditios ude packig stategy fo the geealv(m; ; ) etwoks with ay values We fist taslated the poblem of fidig the ecessay ad sufficiet oblockig coditios fo v(m; ; ) etwoks to a set of liea pogammig poblems We the solved

10 X, X, X, () X, X, X, X, X, X, X, X, X, X, () () () () X, X, X, X, X, X, X, () () (8) (9) X, X, X, X, X, X, X, X, X, () X, () X, X, X, X, X, X, X, X, () (0) X, X, X, () X, X, X, X, X, X, X, X, X, X, () () X, X, X, X, X, X, X, X, X, X, () (8) X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X,+ (9) (0) () Goup (F) Goup (F) Goup (F) Goup (G) Figue : The states of middle switches fo coectio sequece p =(;)(; )(; )(; )(; )(; ) this special type of liea pogammig poblems ad obtaied a elegatclosed fom optimum solutio We have poved that the ecessay ad sufficiet coditio fo a v(m; ; ) etwok to be oblockig ude packig stategy is the umbe of middle switches j, F, k m whee F, is the Fiboacci umbe Fo example, fo = ; ; ad, the ecessay ad sufficiet oblockig coditios ae m, m 9, m,ad m, espectively We believe that the systematic appoach developed i this pape ca be used fo aalyzig othe wide-sese oblockig cotol stategies as well Refeeces [] C Clos, A study of o-blockig switchig etwoks, The Bell System Techical Joual, vol, pp 0-, 9 [] VE Bees, Mathematical Theoy of Coectig Netwoks ad Telephoe Taffic, Academic Pess, New Yok, 9 [] VE Bees, O eaageable thee-stage coectig etwoks, Bell System Techical Joual, vol, No, Septembe 9, pp 8-9 [] A Itoh et al, Pactical implemetatio ad packagig techologies fo a lage-scale ATM switchig system, Joual of Selected Aeas i Commuicatios, vol 9, No 8, pp 80-88, 99 [] J Beetem, M Deeau ad D Weigate, The GF supecompute, Poc of the th Aual Iteatioal Symposium o Compute Achitectue, pp 08-, 98 [] A Jajszczyk ad G Jekel, A ew cocept - epackable etwoks, IEEE Tas Commuicatios, vol, No 8, pp -, 99 [] A Vama ad CS Raghaveda, Itecoectio Netwoks fo Multipocessos ad Multicomputes: Theoy ad Pactice (Tutoial text), IEEE Compute Society Pess, Los Alamitos, CA, 99 [8] MH Ackoyd, Call epackig i coectig etwoks, IEEE Tas Commuicatios, vol, No, pp 89-9, 99 [9] A Giad ad S Hutubise, Dyamic outig ad call epackig i cicuit-switched etwoks, IEEE Tas Commuicatios, vol, No, pp 90-9, 98 [0] Y Mu, Y Tag ad V Devaaja, Aalysis of call packig ad eaagemet i a multi stage switch, IEEE Tas Commuicatios, vol, No //, pp -, 99 [] Y Yag ad J Wag, Wide-seseoblockig Clos etwoks ude packig stategy, Uivesity of Vemot Compute Sciece Reseach Repot, 99 [] DE Kuth, The At of Compute Pogammig, d Editio, vol, Addiso-Wesley Publishig Compay, 9 [] V Chvatal, Liea Pogammig, WH Feema ad Compay, 98 [] KG Muty, Liea Pogammig, Joh Wiley & Sos, 98 [] W Kapla, Advaced Calculus, the Thid Editio, Addiso- Wesley Publishig Compay, 9