Analysis of Supportable Rates in Symmetric Blocking Wavelength Routers

Transcription

1 Aalysis of Supportable Rates i Symmetric Blockig Wavelegth Routers Ca Emre Koksal EPFL School of Computer ad Commuicatio Scieces Lausae, Switzerlad emrekoksal@epflch Abstract Costructig a o-blockig wavelegth router usig optical cross-coects may be impractical due to certai costraits such as the cost or space limitatios Moreover, i may cases the traffic requiremets ca be hadled without a o-blockig router I this paper, we study blockig wavelegth routers costructed usig, < optical crosscoects without wavelegth coversio We fid the set of rates that ca be supported betwee the iput ad output fibers of a certai set of symmetric blockig routers We propose a method to costruct blockig routers to achieve ay give supportable rate i this regio for ay give I INTRODUCTION AND MOTIVATION We study the wavelegth divisio multipleed optical commuicatio system i Fig Each iput ode geerates wavelegths worth of traffic ad each output ode listes o wavelegths There are o wavelegth coverters i the system The traffic patter of the system ca be described by meas of a matri R where R ij is the amout of traffic flowig from iput i to output j Clearly ay coectio matri has to have the property that the sum of each row ad the sum of each colum is less tha or equal to We call a o-blockig if it ca support ay such R Otherwise we call it blockig Figure 2 depicts a oblockig switch (see []) which has a switchig core costructed usig optical crosscoects (OXCs) A OXC is a sigle wavelegth device capable of coectig its iput fibers to the output fibers provided that the coectio patter at ay time is a -to- bipartite matchig The wavelegth router give i Fig 2 ivolves OXCs each with a size Net we cosider usig OXCs of size, < A eample is illustrated i Fig 3 for = 4 ad k = 2 The umber, m = 6, of OXCs i this scheme is larger tha the umber, = 3, of wavelegths at each fiber This meas, each ode has to hadle wavelegths (eg, each ode has trasceivers) as is the case for oblockig routers, whereas there eist m > differet wavelegths i the system O the other had, the size (2 2) of each OXC is cut dow sigificatly Oe ca otice that, i the router give i Fig 3, each iputoutput pair is coected through eactly oe OXC Hece, this router requires that R ij i additio the row ad colum sum coditios Thus, it is blockig iput odes ode wave legths wavelegths ode wavelegth switch output odes ode ode 2 ode Fig Each oe of the odes geerate wavelegth chaels worth of traffic, a fractio of which is destied to some other ode Fig 2 λ optical λ coect λ λ λ optical λ coect λ λ Mu Mu The o-blockig wavelegth router architecture I this paper, we study blockig routers, their fudametal limitatios ad how to build such switches i a systematic way I particular we focus o the architectures whose switchig cores are composed of small OXCs put i parallel There are a umber of motivatios for studyig such routers ) The price of a OXC is proportioal to crosspoit compleity (see [2] for a defiitio) This quatity, o the other had, turs out to be o less tha 2 for stadard architectures

2 2 3 λ λ 2 λ 3 λ 4 Mu Mu Mu 2 3 OXC OXC 2 OXC m Mu Mu λ 5 Fig 4 The blockig router architecture i cosideratio Each ode is coected to of m middle OXCs 4 Mu λ 6 4 Fig 3 A blockig wavelegth router architecture The size of the OXCs are smaller compared to the rearragable versio 2) Eve though o-blockig behavior gives tremedous fleibility to a etwork, it is ot always ecessary I may optical etworks, the amout of traffic betwee ay give two odes is o more tha a few wavelegths 3) The o-blockig wavelegth router give i Fig 2 requires a OXC per wavelegth The spatial dimesios of a OXC grows almost liearly with its size It simply may ot be feasible to stack a OXC at a give lie card due to space limitatios Uder such spatial costraits, smaller OXCs may be preferable There are may variatios of blockig routers, ad i geeral it is ot a trivial task to specify the set of supportable rates for a such routers We itroduce a class of blockig routers ad specify the set of rates that are supporable by the router usig the theory of majorizatio ([3]) We illustrate a systematic way to costruct blockig routers so that these fudametal limits are approached There is a large umber of papers o o-blockig wavelegth routers ad electroic switches (see [4] ad [] for eample) ad blockig cross-coect architectures (see [2] for a review ad [5] for buildig blockig cross-coects usig error cotrol codes) To our kowledge, there is o past work o systematic costructio or aalysis of blockig wavelegth routers m = We assume that the coectios betwee the OXCs ad the output odes are a mirror image of the coectios betwee the OXCs ad the iput odes I a o-blockig router, there is a separate OXC for each wavelegth that a fiber carries ad each iput (ad each output) is coected to every OXC Hece, m = ad = I a blockig router, each iput ad output ode is coected to oly a subset of the OXCs ad vice versa Thus, < < m Compared to a o-blockig architecture the blockig architecture above uses k = / times more OXCs, but each OXC has size smaller by a factor /k Sice the crosspoit compleity of a OXC is superliear i its size, this results i a reductio of crosspoit compleity We represet the supportable rates at a ode with a vector, r of dimesio, ie, r i is the umber of eistig wavelegth coectios betwee a iput ode ad the ith output ode differet from our ode We call a router symmetric if it favors o iput ode or output ode i compariso to others: the set of rates supported by a iput (or output) ode will be idepedet of the ode idetity Also, for a symmetric router, if a give rate vector, r, is supportable, the rp is also supportable for ay permutatio matri P Moreover, by time sharig, ay r ca be supportable so log as it has the form r = π i ( yp i ), () where each P i is a permutatio matri ad the coefficiets π i i Eq costitute a cove combiatio Therefore, if some r is supportable, ay r i the cove hull of r ad ( )! other permutatios of r is also supportable Usig the theorem by Schur (see [3]), all such r are majorized by r, ie, r r II BLOCKING WAVELENGTH ROUTERS A typical wavelegth router ad its parameters are illustrated i Fig 4 (the iteral coectios are ot show i this figure) We deote the umber of odes by, the umber of wavelegths carried at each lik by, the umber of OXCs by m, ad the size of each OXC by It ca be see that Note that we made the iput-output ode separatio for the sake of clarity I reality, there is a uique ode i ad two fibers (oe iput ad oe output) coect this ode to the router Thus, there is o eed for the router to coect iput ode i to output ode i ad the available wavelegths are used to commuicate with the other odes Hece, we defie r to be dimesioal We assume that the iput fiber i ad the output fiber i are coected to the same set of OXCs ad they carry the same set of wavelegths (eve though o OXC coects this pair)

3 I the et sectio, we preset a theorem o subset partitioig, ad based o it we show how to costruct a symmetric router for ay give size of OXCs, Net, for symmetric routers, we show that there eists a maimal vector, w such that if a rate vector, r is supportable by a symmetric router, the it satisfies r w Thus w is the boudary for the rates supportable by ay symmetric router We evaluate w for symmetric routers as a fuctio of, m ad Fially, we prove that a router built usig our costructio supports w Therefore, our costructio achieves all the poits i this capacity regio for supportable rates for symmetric routers III SUPPORTABLE RATES OVER BLOCKING WAVELENGTH ROUTERS I this sectio, we give the mai theorem The we illustrate how we choose the triplet, (,, m) ad how we arrage the iteral coectios to costruct a symmetric router based o of the theorem A Costructio of a Symmetric Router Theorem : Let, m ad be itegers such that m = ( ) There eist subsets of the set {,, m}, such that they all have a cardiality of ( ), ad the itersectio of ay χ, 2 χ of these sets have a cardiality of ( χ χ) The proof is costructive ad ca be foud i [6] Followig is a eample which gives a ituitio o how to costruct such subsets Suppose = 4, = 2 ad m = ( ) = 6 The followig table illustrates the costructio of these 4 subsets of {, 2,, 6} If there is a cross i the (i, j) etry of this table, the the ith subset cotais of a j from the set subset subset 2 subset 3 subset 4 Each elemet of {,, 6} is assiged to every possible 2- combiatio ( = 2) of = 4 sets Each pair (k = 2) of sets has oe ( ( 4 0) ) elemet i commo By costructio, every pair of these subsets has a distict commo elemet i {,, 6} The itersectio of every triple is a empty set Now, suppose we have a system of = 4 odes each with = 3 trasceivers tued to three of m = 6 wavelegths as show i the above table Namely, let the umbers,, 6 (rows) represet the OXCs (distict wavelegths), the 4 subsets (colums) represet the iput (or output) odes ad the crosses represet the iteral coectios betwee iput (ad output) odes ad the OXCs This blockig router is, i fact, the oe illustrated i Fig 3 Sice each pair of odes share eactly oe OXC (ie, wavelegth), each iput ode ca sed up to oe wavelegth of traffic to ay give output ode Ay poit i the regio of rates that are supportable by a ode satisfies the followig: r [ ] Let us go over this costructio for routers with OXCs of size for ay give First, recall that OXC coectios are symmetric for iput ad output odes For istace, if OXC 2 has a coectio to the first iput ode, it is also coected to the first output ode Thus, we shall say, OXC 2 is coected to the first ode, istead of OXC 2 is coected to both the first iput ad the first output ode Sice odes ca be coected to each OXC, each colum of the table cosists of crosses I our costructio we use m = ( ) OXCs Note that this is the ecessary umber of OXCs for symmetry Ideed, if a specific group of iput-output ode pairs share a commo wavelegth, the ay group of iput-output ode pairs must share a commo wavelegth The miimum umber of OXCs eeded for this is m = ( ) (eg, we eed oly m = OXCs if = ) Therefore, our costructio is uique i the sese that, there is o other way to costruct the router ad have symmetry Every combiatio of the odes will be matched with eactly oe of the OXCs (ad a distict wavelegth is assiged to each oe of the m OXCs) Thus, i this costructio each ode carries = m = ( wavelegths, which also equals the umber of OXCs each ode is coected to (eg, each ode carries = wavelegths if = ) Similarly, to fid the umber of OXCs (wavelegths) shared by ay give pair of odes we cout the umber of colums where both of these odes have a cross For such colums, all the 2 combiatios of the other 2 odes are covered, ie, ay give ode pair has ( 2 2) commo wavelegths Also, as a cosequece of Theorem, ay χ, 2 χ odes have ( χ χ) commo wavelegths We ca trivially geeralize this costructio for m = κ ( ), κ Z + such that each oe of the subsets has κ ( ) elemets ad the itersectio of ay give χ of these subsets has a cardiality κ ( χ χ) To achieve this costructio, all we have to do is to repeat the same assigmet κ times for each group of ( ) elemets It ca also be see that the umber of distict wavelegth coectios also icreases by a factor κ with this scalig B Supportable Rates by Symmetric Routers I this sectio we fid a maimal rate vector w which majorizes ay rate r supportable by the costructio The, we show that the router we costructed has the property that each iput or output ode supports ay permutatio of w We start with the symmetric routers with m = ( ) The geeralizatio for the case where κ > will be obvious Let us defie M(η; η ) as the set of OXCs that are commo to iput ode η ad output ode η Also let η η j for j, j ad defie M(η; η, η 2,, η j ) = ) (2) j M(η; η l ) (3) Without loss of geerality, we cosider iput ode What we derive for ode is valid for every ode From basic set l=

4 theory, for ay χ, χ, M(;, 2,, χ) = χ χ M(; j) j= χ j= k=j+ χ 2 χ M(; j, k) + χ j= k=j+ l=k+ M(; j, k, l) For ay symmetric router, for χ <, χ 2 M(;,, χ) = 2 χ 3 χ χ χ, (4) χ χ ad for χ, χ 2 M(;, 2,, χ) = 2 χ χ (5) 0 Let r be a supportable rate vector ad let r = [ r[], r [2],, r [ ] ] deote the decreasig rearragemet of the etries of r Sice M(;,, χ ) is the maimum total umber of full wavelegth coectios that could be set up betwee ay give χ iput-output ode pairs, we ca write the followig set of iequalities for ode r [] M(; ) r [] + r [2] M(;, 2) r [χ] M(;,, ) (6) χ= This set of iequalities costitute a set of upper bouds o the umber of wavelegth coectios that could be set up betwee a iput ode ad the output odes Thus, the vector w, where w χ = M(;,, χ) M(;,, χ ) (7) for χ is a maimal vector for the set of supportable rates, ie, every supportable rate r satisfies r w Net, we show that the router formed by our costructio supports w Iput ode ad output ode ca be coected through o more tha ( 2 2) OXCs M(; ) Let this pair of odes use all these OXCs to set up ( 2 2) coectios, ie, r = M(; ) = ( 2 2) ad thus the first relatio is satisfied with equality Net, startig with χ = 2, we assig OXCs M(;,, χ)\m(;,, χ ) (8) to coect ad χ for χ First, (8) costitutes a valid set of assigmets sice ) The same OXC is ot used to coect a iput ode to two distict output odes, ie, (8) forms disjoit sets for distict χ values 2) The umber of coectios made with this assigmet is idetical to the total umber of wavelegths available at a ode, ie, r χ = M(;,, χ)\m(;,, χ ) χ= χ= ( ) = (9) Sice M(;,, χ ) M(;,, χ), M(;,, χ)\m(;,, χ ) = Therefore, M(;,, χ) M(;,, χ ) w χ = r χ (0) for all χ, ie, (6) is satisfied for all j with equality From symmetry, ay ode ca support ay permutatio of w To summarize, we showed that the vector of umber of wavelegths, r supported by ay give ode η satisfies r w Moreover ay r that satisfies r w is supportable by the symmetric router we costructed Before we fialize the sectio, let us write w eplicitly pluggig Eq (4) ad (5) ito Eq (7) For ay give, ( ) 2 w = 2 w χ = mi{χ, 2} j=0 ( ) j ( χ j )( 2 j 2 j for all χ For eample for = 2, w = [ ] ad for =, w = [( 2) 0 0] Note that for =, w = ad w χ = 0 for all χ > This may mislead oe to the coclusio that the above formulatio fails to calculate the maimal vector for the o-blockig router give i Fig 2 However, for the o-blockig router κ =, hece the maimal vector supportable rates is κ w = [ 0 0 0] Aother remark we would like to make is that w majorizes the vector of umber of wavelegths ot oly from a iput ode but also to a output ode If we deote the umber of wavelegth coectios betwee iput ode i ad output ode j as the (i, j) etry of a rate matri R, the each row ad each colum of R is majorized by w Moreover, ay rate matri whose rows ad colums are majorized by w ca be supported by our symmetric router provided that each row ad colum sum is o more tha ( ) )

5 λ, λ 2 η η 2 λ, λ 3 uidirectioal rig λ, λ 2, λ 3 λ 2, λ 3 Fig 5 A three user uilateral optical rig which supports three wavelegths Every pair of odes has a distict commo wavelegth η 3 IV SUMMARY AND AN EXTENSION I this paper, we preseted a method to costruct a symmetric blockig wavelegth router usig OXCs for ay give < While the crosspoit compleity of this router is ( 2 ) of its o-blockig couterpart, the maimum umber of coectios that could be made betwee ay two odes is reduced by a factor The ideas we developed this paper ca be used for other architectures as well A eample is the rig etwork, which is oe of the most popular architectures for optical etworks Ideed, curretly most of the physical layer ifrastructure is built aroud rigs If a sigle wavelegth is used i a uidirectioal rig, oly oe ode ca achieve full duple commuicatio with aother ode at a time while multiple wavelegths eable may odes to commuicate simultaeously Usig wavelegth multipleers, multiple rigs ca be supported over the same ifrastructure For ay two users to commuicate, they must be able to add ad drop a commo wavelegth Hece, they both eed a add-drop multipleer (ADM) tued to the same wavelegth Suppose we have a costrait that each ode ca have o more tha < ADMs where is the umber of odes We believe that a importat problem is how to tue these ADMs at each ode to achieve certai traffic requiremets If each ode uses the same set of wavelegths, oly a total of wavelegths of traffic ca be supported Now, suppose we use a total of m, m > wavelegths, ad tue each oe of ADMs of every ode similar to that described i Sectio II for blockig wavelegth routers With this modificatio, we icreased the total amout of traffic (i umber of wavelegths) supportable by the rig from to m, without chagig the cost of the etwork (o icrease i umber of ADMs or umber of trasceivers used at each ode) O the other had, the umber of wavelegth coectios possible betwee ay pair of odes is reduced to < As a eample cosider the 3-ode rig etwork i Fig 5 Suppose at most wavelegth of traffic is required betwee each pair of odes We ca tue the ADMs as illustrated i Fig 5 so that a total of m = 3 wavelegth chaels of traffic ca be supported by the etire etwork Each ode ca add ad drop a differet pair of wavelegths so that each pair of odes shares a distict commo wavelegth Thus, at a give time, each ode supports r = [ ] REFERENCES [] Ramaswami R ad Sivarja K N, Optical Networks, Morga Kaufma, 998 [2] Hui J Y, Switchig ad Traffic Theory for Itegrated Broadbad Circuits, Kluwer Academic Publishers, Bosto, MA, 990 [3] Marshall A W ad Olki I, Iequalities: Theory of Majorizatio ad Its Applicatios, Academic Press, New York, NY, 979 [4] Li G H, Noblockig Routig Properties of Clos Networks, i Advaces i Switchig Networks, Bosto, MA, 2000, Kluwer Academic Publishers [5] Koksal C E, A aalysis of blockig switches usig error cotrol codes, i Proceedigs, Iteratioal Symposium o Iformatio Theory, 2004 [6] Masso G M Yag Y, The Necessary Coditios for Clos Type Noblockig Multicast Networks, IEEE Trasactios o Computers, vol 48, o, pp , 999