Multicast Capacity of Optical WDM Packet Ring for Hotspot Traffic
Packet-switching WDM ring networks with a hotspot transporting unicast, multicast, and broadcast traffic are important components of high-speed metropolitan area networks. For an arbitrary multicast fanout traffic model with uniform, hotspot destinat…
Authors: Matthias an der Heiden, Michel Sortais, Michael Scheutzow
MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC MA TTHIAS AN DER HEIDEN, MICHEL SOR T AIS, MICHAEL SCHEUTZO W, MAR TIN REISSLEIN, AND MAR TIN MAIER Abstra t. P a k et-swit hing WDM ring net w orks with a hotsp ot transp orting uniast, m ultiast, and broadast tra are imp ortan t omp onen ts of high-sp eed metrop olitan area net w orks. F or an arbitrary m ultiast fanout tra mo del with uniform, hotsp ot destination, and hotsp ot soure pa k et tra, w e analyze the maxim um a hiev able long-run a v erage pa k et throughput, whi h w e refer to as multi ast ap aity , of bi-diretional shortest-path routed WDM rings. W e iden tify three segmen ts that an exp eriene the maxim um utilization, and th us, limit the m ultiast apait y . W e haraterize the segmen t utilization probabilities through b ounds and appro ximations, whi h w e v er- ify through sim ulations. W e diso v er that shortest-path routing an lead to utilization probabilities ab o v e one half for mo derate to large p ortions of hotsp ot soure m ulti- and broadast tra, and onsequen tly m ultiast apaities of less than t w o sim ultaneous pa k et transmissions. W e outline a one-op y routing strategy that guaran tees a m ultiast apait y of at least t w o sim ultaneous pa k et transmissions for arbitrary hotsp ot soure tra. Keyw ords: Hotsp ot tra, m ultiast, pa k et throughput, shortest path routing, spatial reuse, w a v elength division m ultiplexing (WDM). 1. Intr odution Optial pa k et-swit hed ring w a v elength division m ultiplexing (WDM) net w orks ha v e emerged as a promising solution to alleviate the apait y shortage in the metrop olitan area, whi h is ommonly referred to as metro gap. P a k et-swit hed ring net w orks, su h as the Resilien t P a k et Ring (RPR) [1, 2, 3 ℄, o v erome man y of the shortomings of iruit-swit hed ring net w orks, su h as lo w pro visioning exibilit y for pa k et data tra [4℄. In addition, the use of m ultiple w a v elength hannels in WDM ring net w orks, see e.g., [5 , 6, 7, 8, 9, 10, 11 , 12, 13℄, o v eromes a k ey limitation of RPR, whi h w as originally designed for a single-w a v elength hannel in ea h ring diretion. In optial pa k et-swit hed ring net w orks, the destination no des t ypially remo v e (strip) the pa k ets destined to them from the ring. This destination stripping allo ws the destination no de as w ell as other no des do wnstream to utilize the w a v elength hannel for their o wn transmissions. With this so-alled sp atial wavelength r euse , m ultiple sim ultaneous transmissions an tak e plae on an y giv en w a v elength hannel. Spatial w a v elength reuse is maximized through shortest path routing, whereb y the soure no de sends a pa k et Supp orted b y the DF G Resear h Cen ter Ma theon Mathematis for k ey te hnologies in Berlin. M. an der Heiden, M. Sortais, and M. S heutzo w are with the Departmen t of Mathematis, T e h- nial Univ ersit y Berlin, 10623 Berlin, German y (e-mail: Matthias.an.der.Heidenalumni.TU- Berl in.DE , sortaismath-info.univ-paris5. fr , msmath.tu-berlin.de ). M. Reisslein is with the Dept. of Eletrial Eng., Arizona State Univ., T emp e, AZ 852875706, USA (e-mail: reissleinasu.edu ). M. Maier is with the Institut National de la Re her he Sien tique (INRS), Mon tréal, QC, H5A 1K6, CANAD A (e-mail: maierieee.org ). 1 MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 2 in the ring diretion that rea hes the destination with the smallest hop distane, i.e., tra v ersing the smallest n um b er of in termediate net w ork no des. Multiast tra is widely exp eted to aoun t for a large p ortion of the metro area tra due to m ulti-part y omm uniation appliations, su h as tele-onferenes [ 14 ℄, virtual priv ate net w ork in teronnetions, in terativ e distane learning, distributed games, and on ten t distribution. These m ulti-part y appliations are exp eted to demand substan tial bandwidths due to the trend to deliv er the video omp onen t of m ultimedia on ten t in the High-Denition T elevision (HDTV) format or in video formats with ev en higher resolutions, e.g., for digital inema and tele-immersion appliations. While there is at presen t san t quan titativ e information ab out the m ultiast tra v olume, there is ample anedotal evidene of the emerging signiane of this tra t yp e [15, 16℄. As a result, m ultiasting has b een iden tied as an imp ortan t servie in optial net w orks [17 , 18℄ and has b egun to attrat signian t atten tion in optial net w orking resear h as outlined in Setion 1.1 . Metrop olitan area net w orks onsist t ypially of edge rings that in teronnet sev eral aess net w orks (e.g., Ethernet P assiv e Optial Net w orks) and onnet to a metro ore ring [ 4℄. The metro ore ring in teronnets sev eral metro edge rings and onnets to the wide area net w ork. The no de onneting a metro edge ring to the metro ore ring is t ypially a tra hotsp ot as it ollets/distributes tra destined to/originating from other metro edge rings or the wide area net w ork. Similarly , the no de onneting the metro ore ring to the wide area net w ork is t ypially a tra hotsp ot. Examining the apait y of optial pa k et-swit hed ring net w orks for hotsp ot tra is therefore v ery imp ortan t. In this pap er w e examine the m ultiast apait y (maxim um a hiev able long run a v erage m ulti- ast pa k et throughput) of bidiretional WDM optial ring net w orks with a single hotsp ot for a general fanout tra mo del omprising uniast, m ultiast, and broadast tra. W e onsider an arbitrary tra mix omp osed of uniform tra, hotsp ot destination tra (from regular no des to the hotsp ot), and hotsp ot soure tra (from the hotsp ot to regular no des). W e study the widely onsidered no de ar hiteture that allo ws no des to transmit on all w a v elength hannels, but to reeiv e only on one hannel. W e initially examine shortest path routing b y deriving b ounds and appro xima- tions for the ring segmen t utilization probabilities due to uniform, hotsp ot destination, and hotsp ot soure pa k et tra. W e pro v e that there are three ring segmen ts (in a giv en ring diretion) that go v ern the maxim um segmen t utilization probabilit y . F or the lo kwise diretion in a net w ork with no des 1 , 2 , . . . , N and w a v elengths 1 , 2 , . . . , Λ (with N/ Λ ≥ 1 ), whereb y no de 1 reeiv es on w a v e- length 1, no de 2 on w a v elength 2, . . . , no de Λ on w a v elength Λ , no de Λ + 1 on w a v elength 1, and so on, and with no de N denoting the index of the hotsp ot no de, the three ritial segmen ts are iden tied as: MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 3 ( i ) the segmen t onneting the hotsp ot, no de N , to no de 1 on w a v elength 1, ( ii ) the segmen t onneting no de Λ − 1 to no de Λ on w a v elength Λ , and ( iii ) the segmen t onneting no de N − 1 to no de N on w a v elength Λ . The utilization on these three segmen ts limits the maxim um a hiev able m ultiast pa k et throughput. W e observ e from the deriv ed utilization probabilit y expressions that the utilizations of the rst t w o iden tied segmen ts exeed 1/2 (and approa h 1) for large frations of hotsp ot soure m ulti- and broadast tra, whereas the utilization of the third iden tied segmen t is alw a ys less than or equal to 1/2. Th us, shortest path routing a hiev es a long run a v erage m ultiast throughput of less than t w o sim ultaneous pa k et transmissions (and approa hing one sim ultaneous pa k et transmission) for large p ortions of hotsp ot soure m ulti- and broadast tra. W e sp eify one-op y routing whi h sends only one pa k et op y for hotsp ot soure tra, while uniform and hotsp ot destination pa k et tra is still serv ed using shortest path routing. One-op y routing ensures a apait y of at least t w o sim ultaneous pa k et transmissions for arbitrary hotsp ot soure tra, and at least appro ximately t w o sim ultaneous pa k et transmissions for arbitrary o v er- all tra. W e v erify the auray of our b ounds and appro ximations for the segmen t utilization probabilities, whi h are exat in the limit N/ Λ → ∞ , through omparisons with utilization proba- bilities obtained from disrete ev en t sim ulations. W e also quan tify the gains in maxim um a hiev able m ultiast throughput a hiev ed b y the one-op y routing strategy o v er shortest path routing through sim ulations. This pap er is strutured as follo ws. In the follo wing subsetion, w e review related w ork. In Setion 2, w e in tro due the detailed net w ork and tra mo dels and formally dene the m ultiast apait y . In Setion 3, w e establish fundamen tal prop erties of the ring segmen t utilization in WDM pa k et rings with shortest path routing. In Setion 4, w e deriv e b ounds and appro ximations for the ring segmen t utilization due to uniform, hotsp ot destination, and hotsp ot soure pa k et tra on the w a v elengths that the hotsp ot is not reeiving on, i.e., w a v elengths 1 , 2 , . . . , Λ − 1 in the mo del outlined ab o v e. In Setion 5, w e deriv e similar utilization probabilit y b ounds and appro ximations for w a v elength Λ that the hotsp ot reeiv es on. In Setion 6, w e pro v e that the three sp ei segmen ts iden tied ab o v e go v ern the maxim um segmen t utilization and m ultiast apait y in the net w ork, and disuss impliations for pa k et routing. In Setion 7, w e presen t n umerial results obtained with the deriv ed utilization b ounds and appro ximations and ompare with v erifying sim ulations. W e onlude in Setion 8. 1.1. Related W ork. There has b een inreasing resear h in terest in reen t y ears for the wide range of asp ets of m ultiast in general mesh iruit-swit hed WDM net w orks, inluding ligh tpath design, MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 4 see for instane [19 ℄, tra gro oming, see e.g., [20 ℄, routing and w a v elength assignmen t, see e.g., [21 , 22, 23 ℄, and onnetion arrying apait y [24℄. Similarly , m ultiasting in pa k et-swit hed single-hop star WDM net w orks has b een in tensely in v estigated, see for instane [25 , 26, 27, 28℄. In on trast to these studies, w e fo us on pa k et-swit hed WDM ring net w orks in this pap er. Multiasting in iruit-swit hed WDM rings, whi h are fundamen tally dieren t from the pa k et- swit hed net w orks onsidered in this pap er, has b een extensiv ely examined in the literature. The s heduling of onnetions and ost-eetiv e design of bidiretional WDM rings w as addressed, for instane in [29 ℄. Cost-eetiv e tra gro oming approa hes in WDM rings ha v e b een studied for instane in [30 , 31℄. The routing and w a v elength assignmen t in reongurable bidiretional WDM rings with w a v elength on v erters w as examined in [32 ℄. The w a v elength assignmen t for m ultiasting in iruit-swit hed WDM ring net w orks has b een studied in [33 , 34, 35, 36, 37 , 38℄. F or uniast tra, the throughputs a hiev ed b y dieren t iruit-swit hed and pa k et-swit hed optial ring net w ork ar hitetures are ompared in [39 ℄. Optial p aket-swithe d WDM ring net w orks ha v e b een exp erimen tally demonstrated, see for in- stane [13, 40 ℄, and studied for uniast tra, see for instane [5, 41 , 6 , 7, 8 , 9 , 10 , 11, 12 , 13 ℄. Multiasting in pa k et-swit hed WDM ring net w orks has reeiv ed inreasing in terest in reen t y ears [ 42, 10 ℄. The photonis lev el issues in v olv ed in m ultiasting o v er ring WDM net w orks are explored in [43℄, while a no de ar hiteture suitable for m ultiasting is studied in [44 ℄. The general net w ork ar hiteture and MA C proto ol issues arising from m ultiasting in pa k et-swit hed WDM ring net w orks are addressed in [40 , 45 ℄. The fairness issues arising when transmitting a mix of uniast and m ultiast tra in a ring WDM net w ork are examined in [46℄. The m ultiast apait y of pa k et-swit hed WDM ring net w orks has b een examined for uniform pa k et tra in [47 , 48 , 49, 50℄. In on trast, w e onsider non-uniform tra with a hotsp ot no de, as it ommonly arises in metro edge rings [51 ℄. Studies of non-uniform tra in optial net w orks ha v e generally fo used on issues arising in iruit- swit hed optial net w orks, see for instane [52, 53 , 54, 55 , 56 , 57, 58 ℄. A omparison of iruit-swit hing to optial burst swit hing net w ork te hnologies, inluding a brief omparison for non-uniform tra, w as onduted in [59 ℄. The throughput harateristis of a mesh net w ork in teronneting routers on an optial ring through b er shortuts for non-uniform uniast tra w ere examined in [60℄. The study [61℄ onsidered the throughput harateristis of a ring net w ork with uniform uniast tra, where the no des ma y adjust their send probabilities in a non-uniform manner. The m ultiast apait y of a single-w a v elength pa k et-swit hed ring with non-uniform tra w as examined in [62℄. In on trast to these w orks, w e onsider non-uniform tra with an arbitrary fanout, whi h aommo dates a wide range of uniast, m ultiast, and broadast tra mixes, in a WDM ring net w ork. MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 5 x x x x x x x x x x x x x x x x 1 8 9 13 14 15 N=16 2 3 Λ=4 5 6 7 10 11 12 Figure 2.1. Illustration of the lo kwise w a v elength hannels of a WDM ring net- w ork with N = 16 no des and Λ = 4 w a v elength hannels. 2. System Model and Not a tions W e let N denote the n um b er of net w ork no des, whi h w e index sequen tially b y i, i = 1 , . . . , N , in the lo kwise diretion and let M := { 1 , . . . , N } denote the set of net w ork no des. F or on v eniene, w e lab el the no des mo dulo N , e.g., no de N is also denoted b y 0 or − N . W e onsider the family of no de strutures where ea h no de an transmit on an y w a v elength using either one or m ultiple tunable transmitters ( T T s ) or an arra y of Λ xed-tuned transmitters F T Λ , and reeiv e on one w a v elength using a single xed-tuned reeiv er ( F R ) . F or N = Λ , ea h no de has its o wn home hannel for reeption. F or N > Λ , ea h w a v elength is shared b y η := N / Λ no des, whi h w e assume to b e an in teger. F or 1 ≤ i ≤ N , w e let y u i denote the lo kwise orien ted ring segmen t onneting no de i − 1 to no de i . Analogously , w e let x u i denote the oun ter lo kwise orien ted ring segmen t onneting no de i to no de i − 1 . Ea h ring deplo ys the same set of w a v elength hannels { 1 , . . . , Λ } , one set on the lo kwise ring and another set on the oun terlo kwise ring. The no des n = λ + k Λ with k ∈ { 0 , 1 , . . . , η − 1 } share the drop w a v elength λ . W e refer to the inoming edges of these no des, i.e., the edges y u λ + k Λ and x u λ +1+ k Λ , as riti al e dges on λ . F or m ultiast tra, the sending no de generates a op y of the m ultiast pa k et for ea h w a v elength that is drop w a v elength for at least one destination no de. Denote b y S the no de that is the sender. W e in tro due the random set of destinations (fanout set) F ⊂ ( { 1 , 2 , . . . , N } \ { S } ) . Moreo v er, w e dene the set of ativ e no des A as the union of the sender and all destinations, i.e., A := F ∪ { S } . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 6 W e onsider a tra mo del om bining a p ortion α of uniform tra, a p ortion β of hotsp ot destination tra, and a p ortion γ of hotsp ot soure tra with α, β , γ ≥ 0 and α + β + γ = 1 : Uniform tra: A giv en generated pa k et is a uniform tr a pa k et with probabilit y α . F or su h a pa k et, the sending no de is hosen uniformly at random amongst all net w ork no des { 1 , 2 , . . . , N } . One the sender S is hosen, the n um b er of reeiv ers (fanout) l ∈ { 1 , 2 , . . . , N − 1 } is hosen at random aording to a disrete probabilit y distribution ( µ l ) N − 1 l =1 . One the fanout l is hosen, the random set of destinations (fanout set) F ⊂ ( { 1 , 2 , . . . , N } \ { S } ) is hosen uniformly at random amongst all subsets of { 1 , 2 , . . . , N } \ { S } ha ving ardinalit y l . W e denote b y P α the probabilit y measure asso iated with uniform tra. Hotsp ot destination tra: A giv en pa k et is a hotsp ot destination tr a pa k et with prob- abilit y β . F or a hotsp ot destination tra pa k et, no de N is alw a ys a destination. The sending no de is hosen uniformly at random amongst the other no des { 1 , 2 , . . . , N − 1 } . One the sender S is hosen, the fanout l ∈ { 1 , 2 , . . . , N − 1 } is hosen at random aording to a disrete probabilit y distribution ( ν l ) N − 1 l =1 . One the fanout l is hosen, a random fanout subset F ′ ⊂ ( { 1 , 2 , . . . , N − 1 } \ { S } ) is hosen uniformly at random amongst all subsets of { 1 , 2 , . . . , N − 1 } \ { S } ha ving ardinalit y ( l − 1) , and the fanout set is F = F ′ ∪ { N } . W e denote b y Q β the probabilit y measure asso iated with hotsp ot destination tra. Hotsp ot soure tra: A giv en pa k et is a hotsp ot sour e tr a pa k et with probabilit y γ . F or su h a pa k et, the sending no de is hosen to b e no de N . The fanout 1 ≤ l ≤ ( N − 1) is hosen at random aording to a disrete prob. distribution ( κ l ) N − 1 l =1 . One the fanout l is hosen, a random fanout set F ⊂ { 1 , 2 , . . . , N − 1 } is hosen uniformly at random amongst all subsets of { 1 , 2 , . . . , N − 1 } ha ving ardinalit y l . W e denote b y Q γ the probabilit y measures asso iated with hotsp ot soure tra. While our analysis assumes that the tra t yp e, the soure no de, the fanout, and the fanout set are dra wn indep enden tly at random, this indep endene assumption is not ritial for the analysis. Our results hold also for tra patterns with orrelations, as long as the long run a v erage segmen t utilizations are equiv alen t to the utilizations with the indep endene assumption. F or instane, our results hold for a orrelated tra mo del where a giv en soure no de transmits with a probabilit y p < 1 to exatly the same set of destinations as the previous pa k et sen t b y the no de, and with probabilit y 1 − p to an indep enden tly randomly dra wn n um b er and set of destination no des. W e denote b y P l α the probabilit y measure P α onditioned up on |F | = l , and dene Q l β and Q l γ analogously . W e denote the set of no des with drop w a v elength λ b y (2.1) M λ := { λ + k Λ | k ∈ { 0 , . . . , η − 1 }} . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 7 The set of all destinations with drop w a v elength λ is then (2.2) F λ := F ∩ M λ . Moreo v er, w e use the follo wing notation: F or ℓ ∈ { 1 , . . . , N − 1 } w e denote the probabilit y of ℓ destinations on w a v elength λ b y µ λ,ℓ , ν λ,ℓ , and κ λ,ℓ for uniform, hotsp ot destination, and hotsp ot soure tra, resp etiv ely . Sine the fanout set is hosen uniformly at random among all subsets of { 1 , 2 , . . . , N } \ { S } ha ving ardinalit y l , these usage-probabilities an b e expressed b y µ l , ν l , and κ l . Dep ending on whether the sender is on the drop-w a v elength or not, w e obtain sligh tly dieren t expressions. As will b eome eviden t shortly , it sues to fo us on the ase where the sender is on the onsidered drop w a v elength λ , i.e., S ∈ M λ , sine the relev an t probabilities are estimated through omparisons with transformations (enlarged, redued or righ t/left-shifted ring in tro dued in App endix A) that put the sender in M λ . Through elemen tary om binatorial onsiderations w e obtain the follo wing probabilit y distribu- tions: F or uniform tra, the probabilit y for ha ving ℓ ∈ { 0 , . . . , l ∧ η } destinations on w a v elength λ is (2.3) µ λ,ℓ := N − 1 X l =max(1 ,ℓ ) η ℓ N − η l − ℓ N l µ l . F or hotsp ot destination tra, w e obtain for w a v elengths λ 6 = Λ and ℓ ∈ { 0 , . . . , ( l − 1) ∧ η } (2.4) ν λ,ℓ := N − 1 X l =max(1 ,ℓ ) η ℓ N − η − 1 l − ℓ − 1 N − 1 l − 1 ν l , as w ell as for w a v elength Λ homing the hotsp ot and ℓ ∈ { 1 , . . . , l ∧ η } (2.5) ν Λ ,ℓ := N − 1 X l =1 η − 1 ℓ − 1 N − η l − ℓ N − 1 l − 1 ν l . Finally , for hotsp ot soure tra, w e obtain for λ 6 = Λ and ℓ ∈ { 0 , . . . , l ∧ η } (2.6) κ λ,ℓ := N − 1 X l =max(1 ,ℓ ) η ℓ N − 1 − η l − ℓ N − 1 l κ l and for λ = Λ and ℓ ∈ { 0 , . . . , l ∧ ( η − 1) } (2.7) κ Λ ,ℓ := N − 1 X l =max(1 ,ℓ ) η − 1 ℓ N − η l − ℓ N − 1 l κ l . F or a giv en w a v elength λ , w e denote b y p ℓ α,λ the probabilit y measure P α onditioned up on |F λ | = ℓ , and dene q ℓ β ,λ and q ℓ γ ,λ analogously . R emark 2.1 . Whenev er it is lear whi h w a v elength λ is onsidered w e omit the subsript λ and write p ℓ α , q ℓ β , or q ℓ γ . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 8 W e in tro due the set of ativ e no des A λ on a giv en drop w a v elength λ as (2.8) A λ := F λ ∪ { S } . W e order the no des in this set in inreasing order of their no de indies, i.e., (2.9) A λ = { X λ, 1 , X λ, 2 , . . . , X λ,ℓ +1 } , 1 ≤ X λ, 1 < X λ, 2 < . . . < X λ,ℓ +1 ≤ N , and onsider the gaps (2.10) X λ, 1 + ( N − X λ,ℓ +1 ) , ( X λ, 2 − X λ, 1 ) , . . . , ( X λ,ℓ +1 − X λ,ℓ ) , b et w een suessiv e no des in the set A λ . W e ha v e denoted here again b y ℓ ≡ ℓ λ the random n um b er of destinations with drop w a v elength λ . F or shortest path routing, i.e., to maximize spatial w a v elength reuse, w e determine the largest of these gaps. Sine there ma y b e a tie among the largest gaps (in whi h ase one of the largest gaps is hosen uniformly at random), w e denote the seleted largest gap as C LG λ (for Chosen Largest Gap). Supp ose the C LG λ is b et w een no des X λ,i − 1 and X λ,i . With shortest path routing, the pa k et is then sen t from the sender S to no de X λ,i − 1 , and from the sender S to no de X λ,i in the opp osite diretion. Th us, the largest gap is not tra v ersed b y the pa k et transmission. Note that b y symmetry , P { y u 1 is used } = P { x u N is used } , and P { y u N is used } = P { x u 1 is used } . More generally , for reasons of symmetry , it sues to ompute the utilization probabilities for the lo kwise orien ted edges. F or n ∈ { 1 , . . . , N } , w e abbreviate (2.11) y n λ := y u n is used on w a v elength λ. It will b e on v enien t to all no de N also no de 0 . W e let G λ , G λ = 0 , . . . , N − 1 , b e a random v ariable denoting the rst no de b ordering the hosen largest gap on w a v elength λ , when this gap is onsidered lo kwise. The utilization probabilit y for the lo kwise segmen t n on w a v elength λ is giv en b y (2.12) P y n λ = η X ℓ =0 α · p ℓ α y n λ · µ λ,ℓ + β · q ℓ β y n λ · ν λ,ℓ + γ · q ℓ γ y n λ · κ λ,ℓ . Our primary p erformane metri is the maxim um pa k et throughout (stabilit y limit). More sp eif- ially , w e dene the (eetiv e) m ultiast apait y C M as the maxim um n um b er of pa k ets (with a giv en tra pattern) that an b e sen t sim ultaneously in the long run, and note that C M is giv en as the reipro al of the largest ring segmen t utilization probabilit y , i.e., C M := 1 max n ∈{ 1 ,...,N } max λ ∈{ 1 ,..., Λ } P y n λ . (2.13) MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 9 3. General Pr oper ties of Segment Utiliza tion First, w e pro v e a general reursion form ula for shortest path routing. Prop osition 3.1. L et λ ∈ { 1 , . . . , Λ } b e a xe d wavelength. F or al l no des n ∈ { 0 , . . . , N − 1 } , P y ( n + 1) λ = P y n λ + P ( S = n ) − P ( G λ = n ) . (3.1) Pr o of. There are t w o omplemen tary ev en ts leading to y ( n + 1) λ : (A) the pa k et tra v erses (on w a v e- length λ ) b oth the lo kwise segmen t y u n +1 and the preeding lo kwise segmen t y u n , i.e., the sender is a no de S 6 = n , and (B) no de n is the sender ( S = n ) and transmits the pa k et in the lo kwise diretion, so that it tra v erses segmen t y u n +1 follo wing no de n (in the lo kwise diretion). F ormally , P y ( n + 1) λ = P y n λ and y ( n + 1) λ + P S = n and y ( n + 1) λ . (3.2) Next, note that the ev en t that the lo kwise segmen t y u n is tra v ersed an b e deomp osed in to t w o omplemen tary ev en ts, namely (a) segmen ts y u n and y u n +1 are tra v ersed, and (b) segmen t y u n is tra v ersed, but not segmen t y u n +1 , i.e., P y n λ = P y n λ and y ( n + 1) λ + P y n λ and not y ( n + 1) λ . (3.3) Similarly , w e an deomp ose the ev en t of no de n b eing the sender as (3.4) P ( S = n ) = P S = n and y ( n + 1) λ + P S = n and not y ( n + 1) λ . Hene, w e an express P y ( n + 1) λ as P y ( n + 1) λ = P y n λ − P y n λ and not y ( n + 1) λ + P ( S = n ) − P S = n and not y ( n + 1) λ . (3.5) No w, note that there are t w o omplemen tary ev en ts that result in the CLG to start at no de n , su h that lo kwise segmen t n + 1 is inside the CLG: ( i ) no de n is the last destination no de rea hed b y the lo kwise transmission, i.e., segmen t n is used, but segmen t n + 1 is not used, and ( ii ) no de n is the sender and transmits only a pa k et op y in the oun ter lo kwise diretion. Hene, P ( G λ = n ) = P y n λ and not y ( n + 1) λ + P S = n and not y ( n + 1) λ . (3.6) Therefore, w e obtain the general reursion (3.7) P y ( n + 1) λ = P y n λ + P ( S = n ) − P ( G λ = n ) . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 10 W e in tro due the left (oun ter lo kwise) shift and the righ t (lo kwise) shift of no de n to b e ⌊ n ⌋ λ and ⌈ n ⌉ λ giv en b y (3.8) ⌊ n ⌋ λ := n − λ Λ Λ + λ and ⌈ n ⌉ λ := n − λ Λ Λ + λ. The oun ter lo kwise shift maps a no de n not homed on λ on to the nearest no de in the oun ter lo kwise diretion that is homed on λ . Similarly , the lo kwise shift maps a no de n not homed on λ on to the losest no de in the lo kwise diretion that is homed on λ . F or the tra on w a v elength λ , w e obtain b y rep eated appliation of Prop osition 3.1 P y ( ⌈ n ⌉ λ ) λ = P y n λ + ⌈ n ⌉ λ − 1 X i = n P ( S = i ) − ⌈ n ⌉ λ − 1 X i = n P ( G λ = i ) (3.9) = P y n λ + P ( S ∈ { n , . . . , ⌈ n ⌉ λ − 1 } ) − P ( G λ ∈ { n, . . . , ⌈ n ⌉ λ − 1 } ) . (3.10) Note that the CLG on λ an only start ( i ) at the soure no de, irresp etiv e of whether it is on λ , or ( ii ) at a destination no de on λ . Consider a giv en no de n that is not on λ , then the no des in { n, n + 1 , . . . , ⌈ n ⌉ λ − 1 } are not on λ . (If no de n is on λ , i.e., n = ⌈ n ⌉ λ , then trivially the set { n, n + 1 , . . . , ⌈ n ⌉ λ − 1 } is empt y and P y ( ⌈ n ⌉ λ ) λ = P y n λ .) Hene, the CLG on λ an only start at a no de in { n, n + 1 , . . . , ⌈ n ⌉ λ − 1 } if that no de is the soure no de, i.e., (3.11) P ( G λ ∈ { n, . . . , ⌈ n ⌉ λ − 1 } ) = P ( G λ = S ∈ { n, . . . , ⌈ n ⌉ λ − 1 } ) . Next, note that the ev en t that a no de in { n, n + 1 , . . . , ⌈ n ⌉ λ − 1 } is the soure no de an b e deomp osed in to the t w o omplemen tary ev en ts ( i ) a no de in { n, n + 1 , . . . , ⌈ n ⌉ λ − 1 } is the soure no de and the CLG on λ starts at that no de, and ( ii ) a no de in { n, n + 1 , . . . , ⌈ n ⌉ λ − 1 } is the soure no de and the CLG do es not start at that no de. Hene, (3.12) P ( S ∈ { n, . . . , ⌈ n ⌉ λ − 1 } ) = P ( G λ = S ∈ { n, . . . , ⌈ n ⌉ λ − 1 } ) + P ( S ∈ { n, . . . , m − 1 } , G λ 6 = S ) . Inserting (3.11 ) and (3.12) in (3.10 ) w e obtain (3.13) P y ( ⌈ n ⌉ λ ) λ = P y n λ + P ( S ∈ { n , . . . , m − 1 } , G λ 6 = S ) whi h diretly leads to Corollary 3.2. The usage of non-riti al se gments is smal ler than the usage of riti al se gments, mor e pr e isely for n ∈ { 0 , . . . , N − 1 } : P y n λ = P y ( ⌈ n ⌉ λ ) λ − P ( S ∈ { n , . . . , ⌈ n ⌉ λ − 1 } , G λ 6 = S ) . (3.14) T o ompare the exp eted length of the largest gap on a w a v elength in the WDM ring with the exp eted length of the largest gap in the single w a v elength ring, w e in tro due the enlarged and MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 11 redued ring in App endix A. In brief, in the enlarged ring, an extra no de is added on the onsidered w a v elength b et w een the λ -neigh b ors of the soure no de. This enlargemen t results in ( a ) a set of η + 1 no des homed on the onsidered w a v elength, and ( b ) an enlarged set of ativ e no des A + λ on taining the original destination no des plus the added extra no de (whi h in a sense represen ts the soure no de on the onsidered w a v elength) for a total of ℓ + 1 ativ e no des. The exp eted length of the largest gap on this enlarged w a v elength ring with ℓ + 1 ativ e no des among η + 1 no des homed on the w a v elength (A) is equiv alen t to Λ times the exp eted length of the largest gap on a single w a v elength ring with l = ℓ destination no des and one soure no de among N no des homed on the ring, and (B) pro vides an upp er b ound on the exp eted length of the largest gap on the original w a v elength ring (b efore the enlargemen t). In the redued ring, the left- and righ t-shifted soure no de are merged in to one no de on the onsidered w a v elength, resulting ( a ) in a set of η − 1 no des homed on the onsidered w a v elength, and ( b ) a set A − λ of ℓ − 1 , ℓ , or ℓ + 1 ativ e no des. The exp eted length of the largest gap dereases with inreasing n um b er of ativ e no des, hene w e onsider the ase with ℓ + 1 ativ e no des for a lo w er b ound. The exp eted length of the largest gap on the redued w a v elength ring with ℓ + 1 ativ e no des among η − 1 no des homed on the w a v elength (A) is equiv alen t to Λ times the exp eted length of the largest gap on a single w a v elength ring with l = ℓ destination no des and one soure no de among N no des homed on the ring, and (B) pro vides a lo w er b ound on the exp eted length of the largest gap on the original w a v elength ring (b efore the redution). F rom these t w o onstrutions, whi h are formally pro vided in App endix A, w e diretly obtain: Prop osition 3.3. Given that the ar dinality of F λ is ℓ , the exp e te d length of the CLG on wavelength λ is b ounde d by: (3.15) Λ · g ( ℓ, η − 1) ≤ E ℓ ( | C LG λ | ) ≤ Λ · g ( ℓ, η + 1) , wher e g ( l, N ) denotes the exp e te d length of the CLG for a single wavelength ring with N no des, when the ative set is hosen uniformly at r andom fr om al l subsets of { 1 , . . . , N } with ar dinality ( l + 1) . The exp eted length of the largest gap g ( l, N ) [ 63 ℄ is giv en for l = 0 , . . . , N − 1 , b y g ( l, N ) = P N k =1 k · q l,N ( k ) , where q l,N ( · ) denotes the distribution of the length of the largest gap. Let p l,N ( k ) = N − k − 1 l − 1 / N − 1 l denote the probabilit y that an arbitrary gap has k hops. Then the distribution q l,N ma y b e omputed using the reursion q l,N ( k ) = p l,N ( k ) · k X m =1 q l − 1 ,N − k ( m ) + k − 1 X m =1 p l,N ( m ) · q l − 1 ,N − m ( k ) (3.16) MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 12 together with the initialization q 0 ,N ( k ) = δ N ,k and q N − 1 ,N ( k ) = δ 1 ,k , where δ N ,k denotes the Kro- ne k er Delta. Whereb y , q 0 ,N ( k ) = δ N ,k means a ring with only one ativ e no de has only one gap of length N , hene the largest gap has length N with probabilit y one. Similarly , q N − 1 ,N ( k ) = δ 1 ,k means a ring with all no des ativ e (broadast ase) has N gaps with length one, hene the largest gap has length 1 with probabilit y one. This initialization diretly implies g (0 , N ) = N as w ell as g ( N − 1 , N ) = 1 . Ob viously , w e ha v e to set g ( l, N ) = 0 for l ≥ N . 4. Bounds on Segment Utiliza tion f or λ 6 = Λ 4.1. Uniform T ra. In the setting of uniform tra, one has for all n ∈ {− Λ + λ + 1 , . . . , λ } and k ∈ { 0 , . . . , η − 1 } , for reasons of symmetry: (4.1) P α y n λ = P α y ( n + k Λ) λ . F or n ∈ {− Λ + λ + 1 , . . . , λ } , the dierene b et w een ritial and non-ritial edges, orresp onding to Corollary 3.2, an b e estimated b y 0 ≤ P α ( S ∈ { n, . . . , λ − 1 } , G λ 6 = S ) ≤ P α ( S ∈ { n, . . . , λ − 1 } ) = λ − n N . (4.2) With shortest path routing, on a v erage N − E α ( | C LG | λ ) segmen ts are tra v ersed on λ to serv e a uniform tra pa k et. Equiv alen tly , w e obtain the exp eted n um b er of tra v ersed segmen ts b y sum- ming the utilization probabilities of the individual segmen ts, i.e., as P N n =1 P α y n λ + P N n =1 P α x n λ , whi h, due to symmetry , equals 2 P N n =1 P α y n λ . Hene, N − E α ( | C LG | λ ) = 2 N X n =1 P α y n λ (4.3) and E α ( | C LG | λ ) = N − 2 N X n =1 P α y n λ (4.4) = N − 2 η λ X k = − Λ+ λ +1 P α y k λ . (4.5) Expressing P α y k λ using Corollary 3.2 , w e obtain (4.6) E α ( | C LG | λ ) = N − 2 N P α y λ λ + 2 η λ X k = − Λ+ λ +1 P α ( S ∈ { k , . . . , λ − 1 } , G λ 6 = S ) . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 13 Solving for P α y λ λ yields (4.7) P α y λ λ = 1 2 − 1 2 N E α ( | C LG | λ ) + 1 Λ λ X k = − Λ+ λ +1 P α ( S ∈ { k , . . . , λ − 1 } , G λ 6 = S ) . Hene, the inequalities (4.2) lead to (4.8) 1 2 − 1 2 N E α ( | C LG | λ ) ≤ P α y λ λ ≤ 1 2 − 1 2 N E α ( | C LG | λ ) + Λ − 1 2 N . Emplo ying the b ounds for E α ( | C LG | λ ) from Prop osition 3.3 giv es (4.9) 1 2 − 1 2 η η X ℓ =0 g ( ℓ, η + 1) µ λ,ℓ ≤ P α y λ λ ≤ 1 2 − 1 2 η η − 2 X ℓ =0 g ( ℓ, η − 1) µ λ,ℓ + Λ − 1 2 N . 4.2. Hotsp ot Destination T ra. The only dierene to uniform tra is that N annot b e a sender, sine it is already a destination, i.e., (4.10) q ℓ β y n λ = p ℓ α y n λ | S 6 = N . Using p ℓ α ( S = N ) = 1 N , w e obtain q ℓ β y n λ = N N − 1 p ℓ α y n λ − 1 N − 1 p ℓ α y n λ | S = N (4.11) = N N − 1 p ℓ α y n λ − 1 N − 1 q ℓ γ y n λ . (4.12) Due to the fator 1 N − 1 , the seond term is negligible in the on text of large net w orks. 4.3. Hotsp ot Soure T ra. Sine no de N is the sender (and giv en that there is at least one destination no de on λ ), it sends a pa k et op y o v er segmen t y u n on w a v elength λ if the CLG on λ starts at a no de with index n or higher. Hene, the usage probabilit y of a segmen t an b e omputed as (4.13) q ℓ γ y n λ = q ℓ γ ( G λ ≥ n ) for n ∈ { 1 , . . . , N } . W e notie immediately that q ℓ γ y n λ is monotone dereasing in n . Moreo v er, for all n ∈ { 1 , . . . , ( η − 1) Λ + λ } , Equation (3.14) simplies to q ℓ γ y n λ = q ℓ γ y ( ⌈ n ⌉ λ ) λ (4.14) sine the sender is no de N ≡ 0 and onsequen tly P ( S ∈ { n, . . . , ⌈ n ⌉ λ − 1 } , G λ 6 = S ) = 0 for the onsidered n ∈ { 1 , . . . , ( η − 1) Λ + λ } . Sine q ℓ γ y n λ is monotone dereasing in n , the maximally used ritial segmen t on w a v elength λ is y u λ . With no de N b eing the sender, the CLG on λ an only start at the soure no de N ≡ 0 , or at a destination no de homed on λ . If the CLG do es not start at N ≡ 0 , the segmen t y u λ leading to the MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 14 rst no de homed on λ , namely no de λ , is utilized. Hene, (4.15) q ℓ γ y λ λ = q ℓ γ ( G λ 6 = 0) . Observ e that (4.16) q ℓ γ ( G λ = 0) < q ℓ γ ( G Λ − λ = 0) for λ < Λ 2 , whi h is exploited in Setion 4.4. Enlarging the ring leads to (4.17) q ℓ γ ( G λ = 0) ≤ q ℓ γ G + λ = 0 = 1 ℓ + 1 , sine the gaps b ordering no de 0 are enlarged whereas the lengths of all other gaps are un hanged. A righ t shifting of S yields the follo wing lo w er b ound: q ℓ γ ( G λ = 0) ≥ q ℓ γ ( G → λ = 0 | λ / ∈ F λ ) q ℓ γ ( λ / ∈ F λ ) (4.18) = 1 ℓ + 1 1 − ℓ η . (4.19) Th us, 1 − 1 ℓ + 1 ≤ q ℓ γ y λ λ ≤ 1 − 1 ℓ + 1 1 − ℓ η . (4.20) 4.4. Summary of Segmen t Utilization Bounds and Appro ximation for λ 6 = Λ . F or λ 6 = Λ w e obtain from (2.12 ) and (4.12 ) P y n λ = η X ℓ =0 p ℓ α y n λ αµ λ,ℓ + N N − 1 β ν λ,ℓ + q ℓ γ y n λ γ κ λ,ℓ − 1 N − 1 β ν λ,ℓ . (4.21) Using Corollary 3.2 for p ℓ α and (4.13 ) for q ℓ γ yields (4.22) max n ∈ M P y n λ = P y λ λ , i.e., the segmen t n um b er λ exp erienes the maxim um utilization on w a v elength λ . Moreo v er, in- equalit y (4.16 ) yields (4.23) max λ 6 =Λ max n ∈ M P y n λ = P y 1 1 , i.e., the rst segmen t on w a v elength 1, exp erienes the maxim um utilization among all segmen ts on all w a v elengths λ 6 = Λ . F rom (4.21 ) in onjuntion with (4.9 ) and (4.20 ) w e obtain P y 1 1 ≥ 1 2 α + N N − 1 β − 1 2 η η X ℓ =0 g ( ℓ, η + 1) αµ 1 ,ℓ + N N − 1 β ν 1 ,ℓ + + η X ℓ =0 ℓ ℓ + 1 γ κ 1 ,ℓ − 1 N − 1 β ν 1 ,ℓ =: p 1 l (4.24) MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 15 and P y 1 1 ≤ 1 2 1 + Λ − 1 N α + N N − 1 β − 1 2 η η X ℓ =0 g ( ℓ, η − 1) αµ 1 ,ℓ + N N − 1 β ν 1 ,ℓ + η X ℓ =0 ℓ ( η + 1) ( ℓ + 1) η γ κ 1 ,ℓ − 1 N − 1 β ν 1 ,ℓ =: p 1 u. (4.25) W e obtain an appro ximation of the segmen t utilization b y onsidering the b eha vior of these b ounds for large η = N Λ . Large η imply η +1 η ∼ 1 as w ell as N N − 1 ∼ 1 , and g ( ℓ, η − 1) ∼ g ( ℓ, η + 1) . In tuitiv ely , this last relation means that the exp eted length of the largest gap on a ring net w ork with ℓ destination no des among η − 1 no des is appro ximately equal to the largest gap when there are ℓ destination no des among η + 1 no des. With these onsiderations w e an simplify the b ounds giv en ab o v e and obtain the appro ximation (v alid for large η ): P y 1 1 ∼ 1 2 ( α + β ) − 1 2 η η X ℓ =0 g ( ℓ, η ) ( αµ 1 ,ℓ + β ν 1 ,ℓ ) + γ η X ℓ =0 ℓ ℓ + 1 κ 1 ,ℓ =: p 1 a. (4.26) 5. Bounds on Segment Utiliza tion f or λ = Λ F or uniform tra this ase, of ourse, do es not dier from the ase λ 6 = Λ . 5.1. Hotsp ot Destination T ra. Sine N is a destination no de, b y symmetry it is rea hed b y a lo kwise transmission with probabilit y one half, i.e., (5.1) Q β y N Λ = 1 2 . F or hotsp ot destination tra, no de N an not b e the sender, i.e., Q β ( S = N ) = 0 . Hene, b y Prop osition 3.1 : (5.2) Q β y 1 Λ = 1 2 − Q β ( G Λ = 0) . Moreo v er, w e ha v e from Corollary 3.2 with n = 1 and λ = Λ : (5.3) Q β y Λ Λ = Q β y 1 Λ + Q β ( S ∈ { 1 , . . . , Λ − 1 } , G Λ 6 = S ) . T o estimate Q β ( G Λ = 0) , w e in tro due, as b efore, the left- resp. righ t-shift of S , giv en b y (5.4) ⌊ S ⌋ Λ := S Λ Λ and ⌈ S ⌉ Λ := S Λ Λ . Left and righ t shifting of S leads to the follo wing b ounds for the probabilit y q ℓ β ( G Λ = 0) , whi h are pro v en in App endix B. Prop osition 5.1. F or hotsp ot destination tr a, onditioning on the ar dinality of F Λ to b e ℓ , the pr ob ability that the CLG starts at no de 0 is b ounde d by: (5.5) 1 ℓ + 1 1 − 1 ℓη ≤ q ℓ β ( G Λ = 0) ≤ 1 ℓ + 1 1 + 1 η . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 16 Inserting the b ounds from Prop osition 5.1 and noting that 0 ≤ Q β ( S ∈ { 1 , . . . , Λ − 1 } , G Λ 6 = S ) ≤ (Λ − 1) / (2 N ) leads to (5.6) Q β y Λ Λ ≤ 1 2 − η X ℓ =1 ν Λ ,ℓ 1 ℓ + 1 1 − 1 ℓη + Λ − 1 2 N and (5.7) Q β y Λ Λ ≥ 1 2 − N − 1 X ℓ =1 ν Λ ,ℓ 1 ℓ + 1 1 + 1 η . 5.2. Hotsp ot Soure T ra. Sine w e kno w that N is the sender and has drop w a v elength Λ , w e ha v e a symmetri setting on F Λ and an diretly apply the results of the single w a v elength setting [62℄. In partiular, w e obtain from Setion 3.1.3 in [62 ℄ for ℓ ∈ { 0 , . . . , η − 1 } (5.8) q ℓ γ y N Λ = 0 and (5.9) q ℓ γ y Λ Λ = q ℓ γ ( G Λ 6 = 0) = ℓ ℓ + 1 . 5.3. Summary of Segmen t Utilization Bounds and Appro ximation for λ = Λ . Inserting the b ounds deriv ed in the preeding setions in (2.12 ), w e obtain P y Λ Λ ≥ 1 2 α 1 − 1 η η X ℓ =0 g ( ℓ, η + 1) µ Λ ,ℓ ! + 1 2 β 1 − η X ℓ =1 2( η + 1) ( ℓ + 1) η ν Λ ,ℓ ! + (5.10) + γ η − 1 X ℓ =0 ℓ ℓ + 1 κ Λ ,ℓ =: pLl and P y Λ Λ ≤ 1 2 α 1 + Λ − 1 N − 1 η η X ℓ =0 g ( ℓ, η − 1) µ Λ ,ℓ ! + 1 2 β 1 + Λ − 1 N − η X ℓ =1 2( ℓη − 1) ( ℓ + 1) ℓη ν Λ ,ℓ ! + + γ η − 1 X ℓ =0 ℓ ℓ + 1 κ Λ ,ℓ =: pLu, (5.11) whereb y µ Λ ,ℓ is giv en b y setting λ = Λ in (2.3 ). Moreo v er, P y N Λ ≥ 1 2 α 1 − 1 η η X ℓ =0 g ( ℓ, η + 1) µ Λ ,ℓ ! + 1 2 β =: pN l (5.12) and P y N Λ ≤ 1 2 α 1 + Λ − 1 N − 1 η η X ℓ =0 g ( ℓ, η − 1) µ Λ ,ℓ ! + 1 2 β =: pN u. (5.13) Considering again these b ounds for large η , w e obtain the appro ximations: P y Λ Λ ∼ 1 2 ( α + β ) − α 2 η η X ℓ =0 g ( ℓ, η ) µ Λ ,ℓ − β η X ℓ =1 1 ℓ + 1 ν Λ ,ℓ + γ η − 1 X ℓ =0 ℓ ℓ + 1 κ Λ ,ℓ =: pLa (5.14) MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 17 as w ell as P y N Λ ∼ 1 2 ( α + β ) − α 2 η η X ℓ =0 g ( ℓ, η ) µ Λ ,ℓ =: pN a. (5.15) 6. Ev alua tion of Lar gest Segment Utiliza tion and Seletion of R outing Stra tegy With (4.23 ) and a detailed onsideration of w a v elength λ = Λ , w e pro v e in App endix C the main theoretial result: Theorem 6.1. The maximum se gment utilization pr ob ability is (6.1) max n ∈{ 1 ,...,N } max λ ∈{ 1 ,..., Λ } P y n λ = max P y 1 1 , P y Λ Λ , P y N Λ . It th us remains to ompute the three probabilities on the righ t hand side. W e ha v e no exat result in the most general setting (it w ould b e p ossible to giv e reursiv e form ulae, but these w ould b e prohibitiv ely omplex). Ho w ev er, w e ha v e giv en upp er and lo w er b ounds and appro ximations in Setions 4.4 and 5.3, whi h mat h rather w ell in most situations, as demonstrated in the next setion, and ha v e the same asymptotis when η → ∞ while Λ remains xed. T o w ard assessing the onsidered shortest-path routing strategy , w e diretly observ e, that P y N Λ is alw a ys less or equal to 1 2 . On the other hand, the rst t w o usage probabilities will, for γ large enough, b eome larger than 1 2 , esp eially for hotsp ot soure tra with mo derate to large fanouts. Hene, shortest-path routing will result in a m ultiast apait y of less than t w o for large p ortions of hotsp ot soure m ulti- and broadast tra, whi h ma y arise in on ten t distribution, su h as for IP TV. The in tuitiv e explanation for the high utilization of the segmen ts y 1 1 and y Λ Λ with shortest-path routing for m ulti- and broadast hotsp ot soure tra is a follo ws. Consider the transmission of a giv en hotsp ot soure tra pa k et with destinations on w a v elength Λ homing the hotsp ot. If the pa k et has a single destination uniformly distributed among the other η − 1 no des homed on w a v elength Λ , then the CLG is adjaen t and to the left (i.e., in the oun ter lo kwise sense) of the hotsp ot with probabilit y one half. Hene, with probabilit y one half a pa k et op y is sen t in the lo kwise diretion, utilizing the segmen t y Λ Λ . With an inreasing n um b er of uniformly distributed destination no des on w a v elength Λ , it b eomes less lik ely that the CLG is adjaen t and to the left of the hotsp ot, resulting in inreased utilization of segmen t y Λ Λ . In the extreme ase of a broadast destined from the hotsp ot to all other η − 1 no des homed on Λ , the CLG is adjaen t and to the left of the hotsp ot with probabilit y 1 /η , i.e., segmen t y Λ Λ is utilized with probabilit y 1 − 1 /η . With probabilit y 1 − 2 /η the CLG is not adjaen t to the hotsp ot, resulting in t w o pa k et op y transmissions, i.e., a pa k et op y is sen t in ea h ring diretion. MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 18 F or w a v elength 1, the situation is subtly dieren t due to the rotational oset of the no des homed on w a v elength 1 from the hotsp ot. That is, no de 1 has a hop distane of 1 from the hotsp ot (in the lo kwise diretion), whereas the highest indexed no de on w a v elength 1, namely no de ( η − 1)Λ + 1 has a hop distane of Λ − 1 from the hotsp ot (in the oun ter lo kwise diretion). As for w a v elength Λ , for a giv en pa k et with a single uniformly distributed destination on w a v elength 1, the CLG is adjaen t and to the left of the hotsp ot with probabilit y one half, and the pa k et onsequen tly utilizes segmen t y 1 1 with probabilit y one half. With inreasing n um b er of destinations, the probabilit y of the CLG b eing adjaen t and to the left of the hotsp ot dereases, and the utilization of segmen t y 1 1 inreases, similar to the ase for w a v elength Λ . F or a broadast destined to all η no des on w a v elength 1, the situation is dieren t from w a v elength Λ , in that the CLG is nev er adjaen t to the hotsp ot, i.e., the hotsp ot alw a ys sends t w o pa k et opies, one in ea h ring diretion. 6.1. One-Cop y (OC) Routing. T o o v erome the high utilization of the segmen ts y 1 1 and y Λ Λ due to hotp ot soure m ulti- and broadast tra, w e prop ose one- opy (OC) r outing : With one-op y routing, uniform tra and hotsp ot destination tra are still serv ed using shortest path routing. Hotsp ot soure tra is serv ed using the follo wing oun ter-based p oliy . W e dene the oun ter Y λ to denote the n um b er of no des homed on λ that w ould need to b e tra v ersed to rea h all destinations on λ with one pa k et transmission in the lo kwise diretion (whereb y the nal rea hed destination no de oun ts as a tra v ersed no de). If Y λ < η / 2 , then one pa k et op y is sen t in the lo kwise diretion to rea h all destinations. If Y λ > η / 2 , then one pa k et op y is sen t in the oun ter lo kwise diretion to rea h all destinations. Ties, i.e., Y λ = η / 2 , are serv ed in either lo kwise or oun ter lo kwise diretion with probabilit y one half. F or hotsp ot soure tra with arbitrary tra fanout, this oun ter-based one-op y routing ensures a maxim um utilization of one half on an y ring segmen t. Note that the oun ter-based p oliy onsiders only the no des homed on the onsidered w a v elength λ to ensure that the rotational oset b et w een the w a v elength Λ homing the hotsp ot and the onsidered w a v elength λ do es not aet the routing deisions. W e prop ose the follo wing strategy for swit hing b et w een shortest path (SP) and one-op y (OC) routing. Shortest path routing is emplo y ed if b oth (4.26 ) and (5.14 ) are less than one half. If (4.26 ) or (5.14 ) exeeds one half, then one-op y routing is used. F or the pratial implemen tation of this swit hing strategy , the hotsp ot an p erio dially estimate the urren t tra parameters, i.e., the tra p ortions α , β , and γ as w ell as the orresp onding fanout distributions µ l , ν l , and κ l , l = 1 , . . . , N − 1 , for instane, through a om bination of tra measuremen ts and histori tra patterns, similar to [64 , 65, 66, 67, 68℄. F rom these tra parameter estimates, the hotsp ot an then ev aluate (4.26 ) and (5.14 ). MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 19 T o obtain a more rened riterion for swit hing b et w een shortest path routing and one-op y routing w e pro eed as follo ws. W e haraterize the maxim um segmen t utilization with shortest path routing more expliitly b y inserting (4.26 ), (5.14 ), and (5.15 ) in (6.2 ) to obtain: max n ∈{ 1 ,...,N } max λ ∈{ 1 ,..., Λ } P y n λ = 1 2 ( α + β ) − α 2 η η X ℓ =0 g ( ℓ, η ) µ 1 ,ℓ + max ( 0 , − β 2 η η X ℓ =0 g ( ℓ, η ) ν 1 ,ℓ + γ η X ℓ =0 ℓ ℓ + 1 κ 1 ,ℓ , − β η X ℓ =1 1 ℓ + 1 ν Λ ,ℓ + γ η − 1 X ℓ =0 ℓ ℓ + 1 κ Λ ,ℓ ) , (6.2) whereb y w e noted that the denition of µ λ,ℓ in (2.3 ) diretly implies that µ λ,ℓ is indep enden t of λ . Clearly , the hotsp ot soure tra do es not inuene the maxim um segmen t utilization as long as γ ≤ γ th 1 , 1 := β 2 η P η ℓ =0 g ( ℓ, η ) ν 1 ,ℓ P η ℓ =1 ℓ ℓ +1 κ 1 ,ℓ (6.3) and γ ≤ γ th 1 , Λ := β P η ℓ =1 1 ℓ +1 ν Λ ,ℓ P η − 1 ℓ =1 ℓ ℓ +1 κ Λ ,ℓ . (6.4) Th us, if γ ≤ γ th 1 = min( γ th 1 , 1 , γ th 1 , Λ ) , then all tra is serv ed using shortest path routing. W e next note that Theorem 6.1 do es not hold for the one-op y routing strategy . W e therefore b ound the maxim um segmen t utilization probabilit y with one-op y routing b y observing that (4.9 ) together with Prop osition 3.2 and (4.2 ) implies that asymptotially for all λ ∈ { 1 , . . . , Λ } (6.5) P α y n λ ∼ 1 2 − 1 2 η η − 1 X ℓ =0 g ( ℓ, η ) µ λ,ℓ . Hene, P α y n λ is asymptotially onstan t. Moreo v er, similar as in the single w a v elength ase [ 62 ℄, w e ha v e (6.6) P β y n λ ≤ P β y N Λ = 1 2 . Therefore, the maxim um segmen t utilization with one-op y routing is (appro ximately) b ounded b y max n ∈{ 1 ,...,N } max λ ∈{ 1 ,..., Λ } P y n λ ≤ 1 2 ( α + β + γ ) − α 2 η η − 1 X ℓ =0 g ( ℓ, η ) µ 1 ,ℓ . (6.7) Comparing (6.7 ) with (6.2 ) w e observ e that the maxim um segmen t utilization with one-op y routing is smaller than with shortest path routing if the follo wing threshold onditions hold: • If P η ℓ =1 ℓ ℓ +1 κ 1 ,ℓ > 1 2 , then set γ th 2 , 1 = β 2 η P η ℓ =0 g ( ℓ, η ) ν 1 ,ℓ P η ℓ =1 ℓ ℓ +1 κ 1 ,ℓ − 1 2 , (6.8) otherwise set γ th 2 , 1 = ∞ . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 20 • If P η − 1 ℓ =1 ℓ ℓ +1 κ Λ ,ℓ > 1 2 , then set γ th 2 , Λ := β P η ℓ =1 1 ℓ +1 ν Λ ,ℓ P η − 1 ℓ =1 ℓ ℓ +1 κ Λ ,ℓ − 1 2 , (6.9) otherwise set γ th 2 , Λ = ∞ . If γ ≥ γ th 2 = max( γ th 2 , 1 , γ th 2 , Λ ) , then one-op y routing is emplo y ed. F or γ v alues b et w een γ th 1 and γ th 2 , the hotsp ot ould n umerially ev aluate the maxim um seg- men t utilization probabilit y of shortest path routing with the deriv ed appro ximations. The hotsp ot ould also obtain the segmen t utilization probabilities with one-op y routing through disrete ev en t sim ulations to determine whether shortest path routing or one-op y routing of the hotsp ot tra is preferable for a giv en set of tra parameter estimates. 7. Numerial and Simula tion Resul ts In this setion w e presen t n umerial results obtained from the deriv ed b ounds and appro ximations of the utilization probabilities as w ell as v erifying sim ulations. W e initially sim ulate individual, sto hastially indep enden t pa k ets generated aording to the tra mo del of Setion 2 and routed aording to the shortest path routing p oliy . W e determine estimates of the utilization probabilities of the three segmen ts y 1 1 , y Λ Λ , and y N Λ and denote these probabilities b y p 1 s , pLs , and pN s . Ea h sim ulation is run un til the 99% ondene in terv als of the utilization probabilit y estimates are less than 1% of the orresp onding sample means. W e onsider a net w orks with Λ = 4 w a v elength hannels in ea h ring diretion. 7.1. Ev aluation of Segmen t Utilization Probabilit y Bounds and Appro ximations for Shortest P ath Routing. W e examine the auray of the deriv ed b ounds and appro ximations b y plotting the segmen t utilization probabilities as a funtion of the n um b er of net w ork no des N = 8 , 12 , 16 , . . . , 256 and omparing with the orresp onding sim ulation results. F or the rst set of ev aluations, w e onsider m ultiast tra with xed fanout µ 1 = ν 1 = κ 1 = 1 / 4 and µ l = ν l = κ l = 3 / (4 ( N − 2)) for l = 2 , . . . , N − 1 . W e examine inreasing p ortions of hotsp ot tra b y setting α = 1 , β = γ = 0 for Fig. 7.1, α = 0 . 6 , β = 0 . 1 , and γ = 0 . 3 for Fig. 7.2, and α = 0 . 2 , β = 0 . 2 , and γ = 0 . 6 for Fig. 7.3. W e onsider these senarios with hotsp ot tra dom- inated b y hotsp ot soure tra, i.e., with γ > β , sine man y m ultiast appliations in v olv e tra distribution b y a hotsp ot, e.g., for IP TV. W e also onsider a xed tra mix α = 0 . 2 , β = 0 . 2 , and γ = 0 . 6 for inreasing fanout. W e onsider uniast (UC) tra with µ 1 = ν 1 = κ 1 = 1 in Fig. 7.4, mixed tra (MI) with µ 1 = ν 1 = κ 1 = 1 / 2 and µ l = ν l = κ l = 1 / (2( N − 2)) for l = 2 , . . . , N − 1 in Fig. 7.5, m ultiast (MC) tra MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 21 0 0.1 0.2 0.3 0.4 0.5 0.6 0 50 100 150 200 250 Utilization Probability Number of Nodes p1u p1s p1a p1l 0 0.1 0.2 0.3 0.4 0.5 0.6 0 50 100 150 200 250 Utilization Probability Number of Nodes pLu pLs pLa pLl 0 0.1 0.2 0.3 0.4 0.5 0.6 0 50 100 150 200 250 Utilization Probability Number of Nodes pNu pNs pNa pNl (a) P y 1 1 (b) P y 4 4 () P y 64 4 Figure 7.1. Segmen t utilization probabilit y as a funtion of n um b er of No des N for α = 1 , β = 0 , γ = 0 , and µ 1 = ν 1 = κ 1 = 1 / 4 and µ l = ν l = κ l = 3 / (4( N − 2)) for l = 2 , . . . , N − 1 . 0.1 0.2 0.3 0.4 0.5 0.6 0 50 100 150 200 250 Utilization Probability Number of Nodes p1u p1s p1a p1l 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 50 100 150 200 250 Utilization Probability Number of Nodes pLu pLs pLa pLl 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 50 100 150 200 250 Utilization Probability Number of Nodes pNu pNs pNa pNl (a) P y 1 1 (b) P y 4 4 () P y 64 4 Figure 7.2. Segmen t utilization probabilit y as a funtion of n um b er of No des N for α = 0 . 6 , β = 0 . 1 , γ = 0 . 3 , and µ 1 = ν 1 = κ 1 = 1 / 4 and µ l = ν l = κ l = 3 / (4( N − 2)) for l = 2 , . . . , N − 1 . 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 50 100 150 200 250 Utilization Probability Number of Nodes p1u p1s p1a p1l 0.1 0.2 0.3 0.4 0.5 0.6 0 50 100 150 200 250 Utilization Probability Number of Nodes pLu pLs pLa pLl 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0 50 100 150 200 250 Utilization Probability Number of Nodes pNu pNs pNa pNl (a) P y 1 1 (b) P y 4 4 () P y 64 4 Figure 7.3. Segmen t utilization probabilit y as a funtion of n um b er of No des N for α = 0 . 2 , β = 0 . 2 , γ = 0 . 6 , and µ 1 = ν 1 = κ 1 = 1 / 4 and µ l = ν l = κ l = 3 / (4( N − 2)) for l = 2 , . . . , N − 1 . with µ l = ν l = κ l = 1 / ( N − 1) for l = 1 , . . . , N − 1 in Fig. 7.6 , and broadast (BC) tra with µ N − 1 = ν N − 1 = κ N − 1 = 1 in Fig. 7.7. MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 22 0 0.05 0.1 0.15 0.2 0.25 0.3 0 50 100 150 200 250 Utilization Probability Number of Nodes p1u p1s p1a p1l 0 0.05 0.1 0.15 0.2 0.25 0 50 100 150 200 250 Utilization Probability Number of Nodes pLu pLs pLa pLl 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 50 100 150 200 250 Utilization Probability Number of Nodes pNu pNs pNa pNl (a) P y 1 1 (b) P y 4 4 () P y 64 4 Figure 7.4. Segmen t utilization probabilit y as a funtion of n um b er of No des N for α = 0 . 2 , β = 0 . 2 , γ = 0 . 6 , and uniast (UC) tra with µ 1 = ν 1 = κ 1 = 1 . 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 50 100 150 200 250 Utilization Probability Number of Nodes p1u p1s p1a p1l 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 50 100 150 200 250 Utilization Probability Number of Nodes pLu pLs pLa pLl 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0 50 100 150 200 250 Utilization Probability Number of Nodes pNu pNs pNa pNl (a) P y 1 1 (b) P y 4 4 () P y 64 4 Figure 7.5. Segmen t utilization probabilit y as a funtion of n um b er of No des N for α = 0 . 2 , β = 0 . 2 , γ = 0 . 6 , for mixed (MI) tra with µ 1 = ν 1 = κ 1 = 1 / 2 and µ l = ν l = κ l = 1 / (2( N − 2)) for l = 2 , . . . , N − 1 . 0.3 0.4 0.5 0.6 0.7 0 50 100 150 200 250 Utilization Probability Number of Nodes p1u p1s p1a p1l 0.2 0.3 0.4 0.5 0.6 0.7 0 50 100 150 200 250 Utilization Probability Number of Nodes pLu pLs pLa pLl 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0 50 100 150 200 250 Utilization Probability Number of Nodes pNu pNs pNa pNl (a) P y 1 1 (b) P y 4 4 () P y 64 4 Figure 7.6. Segmen t utilization probabilit y as a funtion of n um b er of No des N for α = 0 . 2 , β = 0 . 2 , γ = 0 . 6 , for m ultiast (MC) tra with µ l = ν l = κ l = 1 / ( N − 1) for l = 1 , . . . , N − 1 . W e observ e from these gures that the b ounds get tigh t for mo derate to large n um b ers of no des N and that the appro ximations haraterize the atual utilization probabilities fairly aurately for the full range of N . F or instane, for N = 64 no des, the dierene b et w een the upp er and lo w er MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 23 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0 50 100 150 200 250 Utilization Probability Number of Nodes p1u p1s p1a p1l 0.3 0.4 0.5 0.6 0.7 0.8 0 50 100 150 200 250 Utilization Probability Number of Nodes pLu pLs pLa pLl 0.14 0.16 0.18 0.2 0.22 0.24 0 50 100 150 200 250 Utilization Probability Number of Nodes pNu pNs pNa pNl (a) P y 1 1 (b) P y 4 4 () P y 64 4 Figure 7.7. Segmen t utilization probabilit y as a funtion of n um b er of No des N for α = 0 . 2 , β = 0 . 2 , γ = 0 . 6 , for broadast (BC) tra with µ N − 1 = ν N − 1 = κ N − 1 = 1 . b ound is less than 0.06, for N = 128 this dierene shrinks to less than 0.026. The magnitudes of the dierenes b et w een the utilization probabilities obtained with the analytial appro ximations and the atual sim ulated utilization probabilities are less than 0.035 for N = 64 no des and less than 0.019 for N = 128 for the wide range of senarios onsidered in Figs. 7.1 7.7 . (When exluding the broadast ase onsidered in Fig. 7.7 , these magnitude dierenes shrink to 0.02 for N = 64 no des and 0.01 for N = 128 no des.) F or some senarios w e observ e for small n um b er of no des N sligh t osillations of the atual utiliza- tion probabilities obtained through sim ulations, e.g., in Fig. 7.4(a) and 7.5 (a). More sp eially , w e observ e p eaks of the utilization probabilities for o dd η and v alleys for ev en η . These osillations are due to the disrete v ariations in the n um b er of destination no des leading to segmen t tra v ersals. F or instane, for the hotsp ot soure uniast tra that aoun ts for a γ = 0 . 6 p ortion of the tra in Fig. 7.4 (a), the utilization of segmen t y 1 1 is as follo ws. F or ev en η , there are η / 2 p ossible destination no des that result in tra v ersal of segmen t y 1 1 , ea h of these destination no des o urs with probabilit y 1 / ( N − 1) ; hene, segmen t y 1 1 is tra v ersed with probabilit y N/ [2Λ( N − 1)] . On the other hand, for o dd η , there are ( η + 1) / 2 p ossible destination no des that result in tra v ersal of segmen t y 1 1 ; hene, segmen t y 1 1 is tra v ersed with probabilit y ( N + Λ) / [2 Λ( N − 1)] . Ov erall, w e observ e from Fig 7.1 that for uniform tra, the three segmen ts go v erning the max- im um utilization probabilit y are ev enly loaded. With inreasing frations of non-uniform tra (with hotsp ot soure tra dominating o v er hotsp ot destination tra), the segmen ts y 1 1 and y 4 4 exp eriene inreasing utilization probabilities ompared to segmen t y 64 4 , as observ ed in Figs. 7.2 and 7.3 . Similarly , for the non-uniform tra senarios with dominating hotsp ot soure tra, w e observ e from Figs. 7.47.7 inreasing utilization probabilities for the segmen ts y 1 1 and y 4 4 ompared to segmen t y 64 4 with inreasing fanout. (In senarios with dominating hotsp ot destination tra, not MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 24 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Max Util Prob gamma BC, SP BC, OC MC, SP MC, OC MI, SP MI, OC UC, SP UC, OC 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Max Util Prob gamma BC, SP BC, OC MC, SP MC, OC MI, SP MI, OC UC, SP UC, OC (a) β = 0 . 1 (b) β = 0 . 2 Figure 7.8. Maxim um segmen t utilization probabilit y as a funtion of fration of hotsp ot soure tra γ (with α = 1 − β − γ ) for shortest path (SP) and one-op y routing (OC) for xed fration of hotsp ot tra β for uniast (UC) tra, mixed (MI) tra, m ultiast (MC) tra, and broadast (BC) tra. sho wn here due to spae onstrain ts, the utilization of segmen t y 64 4 inreases ompared to segmen ts y 1 1 and y 4 4 .) In Figs. 7.3, 7.6, and 7.7 , the utilization probabilities for segmen ts y 1 1 and y 4 4 exeed one half for senarios with mo derate to large n um b ers of no des (and orresp ondingly large fanouts), indiating the p oten tial inrease in m ultiast apait y b y emplo ying one-op y routing. 7.2. Comparison of Segmen t Utilization Probabilities for SP and OC Routing. In Fig. 7.8 w e ompare shortest path routing (SP) with one-op y routing (OC) for uniast (UC) tra, mixed (MI) tra, m ultiast (MC) tra, and broadast (BC) tra with the fanout distributions dened ab o v e for a net w ork with N = 128 no des. The orresp onding thresholds γ th 1 and γ th 2 are rep orted in T able 1. F or SP routing, w e plot the maxim um segmen t utilization probabilit y obtained from the analytial appro ximations. F or OC routing, w e estimate the utilization probabilities of all segmen ts in the net w ork through sim ulations and then sear h for the largest segmen t utilization probabilit y . F o using initially on uniast tra, w e observ e that b oth SP and OC routing attain the same maxim um utilization probabilities. This is to b e exp eted sine the routing b eha viors of SP and OC are iden tial when there is a single destination on a w a v elength. F or β = 0 . 1 , w e observ e with inreasing p ortion of hotsp ot soure tra γ an initial derease, a minim um v alue, and subsequen t inrease of the maxim um utilization probabilit y . The v alue of the maxim um utilization probabilit y for γ = 0 is due to the uniform and hotsp ot destination tra hea vily loading segmen t y 64 4 . With inreasing γ and onsequen tly dereasing α , the load on segmen t y 64 4 diminishes, while the load on segmen ts y 1 1 and y 4 4 inreases. F or appro ximately γ = 0 . 4 , the three segmen ts y 1 1 , y 4 4 , and y 64 4 are MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 25 F anout γ th 1 γ th 2 β = 0 . 1 UC 0.397 ∞ MI 0.059 7.32 MC 0.011 0.030 BC 0.0004 0.006 β = 0 . 2 UC 0.794 ∞ MI 0.118 14.64 MC 0.022 0.061 BC 0.0008 0.013 T able 1. Thresholds γ th 1 and γ th 2 for senarios onsidered in Fig. 7.8 ab out equally loaded. As γ inreases further, the segmen ts y 1 1 and y 4 4 exp eriene roughly the same, inreasing load. F or β = 0 . 2 w e observ e only the derease of the maxim um utilization probabilit y , whi h is due to the load on segmen t y 64 4 dominating the maxim um segmen t utilization. F or this larger fration of hotsp ot destination tra w e do not rea h the regime where segmen ts y 1 1 and y 4 4 go v ern the maxim um segmen t utilization. T urning to broadast tra, w e observ e that SP routing giv es higher maxim um utilization proba- bilities than OC routing for essen tially the en tire range of γ , rea hing utilization probabilities around 0.9 for high prop ortions of hotsp ot soure tra. This is due to the high loading of segmen ts y 1 1 and y 4 4 . In on trast, with OC routing, the maxim um segmen t utilization sta ys lose to 0.5, resulting in signian tly inreased apait y . The sligh t exursions of the maxim um OC segmen t utilization probabilit y ab o v e 1/2 are due to uniform tra. The segmen t utilization probabilit y with uniform tra is appro ximated (not b ounded) b y (6.5), making exursions ab o v e 1/2 p ossible ev en though hotsp ot destination and hotsp ot soure tra result in utilization probabilities less than (or equal) to 1/2. F or mixed and m ultiast tra, w e observ e for inreasing γ an initial derease, minim um v alue, and subsequen t inrease of the maxim um utilization probabilit y for b oth SP and OC routing. Similarly to the ase of uniast tra, these dynamis are aused b y initially dominating loading of segmen t y 64 4 , then a derease of the loading of segmen t y 64 4 while the loads on segmen ts y 1 1 and y 4 4 inrease. W e observ e for the mixed and m ultiast tra senarios with the same fanout for all three tra t yp es onsidered in Fig 7.8 that SP routing and OC routing giv e essen tially the same maxim um segmen t utilization for small γ up to a knee p oin t in the SP urv es. F or larger γ , OC routing giv es signian tly smaller maxim um segmen t utilizations. W e observ e from T able 1 that for relativ ely large fanouts (MC and BC), the ranges b et w een γ th 1 and γ th 2 are relativ ely small, limiting the need for resorting to n umerial ev aluation and sim ulation for determining whether to emplo y SP or OC MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 26 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Max Util Prob gamma SP,d=127 OC,d=127 SP,d=64 OC,d=64 SP,d=1 OC,d=1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Max Util Prob gamma SP,d=127 OC,d=127 SP,d=64 OC,d=64 SP,d=1 OC,d=1 (a) ν 8 = 1 , κ d = 1 (b) ν d = 1 , κ 64 = 1 Figure 7.9. Maxim um segmen t utilization probabilit y as a funtion of fration of hotsp ot soure tra γ . Fixed parameters: N = 128 no des, β = 0 . 4 , µ l = 1 / 16 for l = 1 , . . . , 16 . Senario γ th 1 γ th 2 κ d = 1 d = 127 0.122 0.283 d = 64 0.126 0.302 d = 1 0.972 ∞ ν d = 1 d = 127 0.0017 0.028 d = 64 0.025 0.073 d = 1 0.212 0.456 T able 2. Thresholds γ th 1 and γ th 2 for senarios onsidered in Fig. 7.9 routing. F or small fanouts (UC and MI), the γ thresholds are far apart; further rened deision riteria for routing with SP or OC are therefore an imp ortan t diretion for future resear h. W e ompare shortest path (SP) and one-op y (OC) routing for senarios with dieren t fanout distribution for the dieren t tra t yp es in Fig. 7.9 for a ring with N = 128 no des. W e observ e from Fig. 7.9 (a) that for hotsp ot soure tra with large fanout, SP routing a hiev es signian tly smaller maxim um segmen t utilizations than OC routing for γ v alues up to a ross-o v er p oin t, whi h lies b et w een γ th 1 and γ th 2 . Similarly , w e observ e from Fig. 7.9 (b) that for small γ , SP routing a hiev es signian tly smaller maxim um segmen t utilizations than OC routing for hotsp ot destination tra with small fanout. F or example, for uniast hotsp ot destination tra (i.e., ν 1 = 1 ), for γ = 0 . 21 , SP routing giv es a m ultiast apait y of C M = 3 . 72 ompared to C M = 3 . 19 with OC routing. By swit hing from SP routing to OC routing when the fration of hotsp ot soure tra γ exeeds 0.31, the smaller maxim um utilization probabilit y , i.e., higher m ultiast apait y an b e a hiev ed aross the range of frations of hotsp ot soure tra γ . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 27 8. Conlusion W e ha v e analytially haraterized the segmen t utilization probabilities in a bi-diretional WDM pa k et ring net w ork with a single hotsp ot. W e ha v e onsidered arbitrary mixes of uniast, m ultiast, and broadast tra in om bination with an arbitrary mix of uniform, hotsp ot destination, and hotsp ot soure tra. F or shortest-path routing, w e found that there are three segmen ts that an attain the maxim um utilization, whi h in turn limits the maxim um a hiev able long-run a v erage m ultiast pa k et throughput (m ultiast apait y). Through v erifying sim ulations, w e found that our b ounds and appro ximations of the segmen t utilization probabilities, whi h are exat in the limit for man y no des in a net w ork with a xed n um b er of w a v elength hannels, are fairly aurate for net w orks with on the order of ten no des reeiving on a w a v elength. Imp ortan tly , w e observ ed from our segmen t utilization analysis that shortest-path routing do es not maximize the a hiev able m ultiast pa k et throughput when there is a signian t p ortion of m ulti- or broadast tra emanating from the hotsp ot, as arises with m ultimedia distribution, su h as IP TV net w orks. W e prop osed a one-op y routing strategy with an a hiev able long run a v erage m ultiast pa k et throughout of ab out t w o sim ultaneous pa k et transmissions for su h distribution senarios. This study fo used on the maxim um a hiev able m ultiast pa k et throughput, but did not on- sider pa k et dela y . A thorough study of the pa k et dela y in WDM ring net w orks with a hotsp ot transp orting m ultiast tra is an imp ortan t diretion for future resear h. Appendix A. Definition of Enlar ged and Redued Ring as well as of Left ( A ← λ ) and Right Shifting ( A → λ ) of Set of A tive Nodes In this app endix, w e rst dene the enlarging and reduing of the set of " λ -ativ e no des" A λ := F λ ∪ { S } . Supp ose that |F λ | = ℓ . Dep ending on the setting, and with M λ denoting the set of no des homed on a giv en w a v elength λ , the set F λ is hosen uniformly at random among • all subsets of M λ (uniform tra and for λ 6 = Λ also hotsp ot destination and soure tra), or • all subsets of M λ that on tain N (hotsp ot destination tra for λ = Λ sine N is alw a ys a destination for hotsp ot destination tra), or • all subsets of M λ that do not on tain N (hotsp ot soure tra for λ = Λ sine N is alw a ys the soure for hotsp ot soure tra). Assuming S / ∈ M λ , w e dene: MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 28 enlarged ring: W e enlarge the set M λ b y injeting an extra no de homed on λ b et w een ⌊ S ⌋ λ and ⌈ S ⌉ λ (and orresp ondingly Λ − 1 no des homed on the other w a v elengths). After a re- n umeration starting with 0 at the new no de (whi h is aordingly homed on w a v elength Λ after the re-n umeration), w e obtain M Λ ,η + 1 := m Λ m ∈ { 0 , . . . , η } . W e dene the enlarged set F + λ to equal the ren um b ered set F λ united with the new no de. This pro edure leads to a random set of ativ e no des A + λ = F + λ that is uniformly distributed among all subsets of M Λ ,η + 1 with ardinalit y ( ℓ + 1) on taining no de 0 . Note that the largest gap of the enlarged set is larger or equal to the largest gap of A λ . 4 =0 X 3 X X 1 X 2 S 3 X enlarge 1 X 2 X = =2 =3 4 Λ Λ Λ Λ Figure A.1. Example of enlarging M 3 for N = 16 , Λ = 4 . The sender homed on w a v elength 1 is represen ted b y S in the left illustration. The no des of M 3 are indiated b y longer ti k marks and the no des of F 3 are irled. The enlarged ring has a total of N + Λ = 20 no des, with η + 1 = 5 no des homed on ea h w a v elength. The added no de on w a v elength 3 is n um b ered with 0 and lies b et w een the former ⌊ S ⌋ λ and ⌈ S ⌉ λ . redued ring: W e transform the set M λ b y merging the no des ⌊ S ⌋ λ and ⌈ S ⌉ λ to a single ativ e no de (eliminating the Λ − 1 no des in b et w een). After re-n umeration starting with 0 at this merged no de, w e obtain an ativ e set A − λ on M Λ ,η − 1 . Dep ending on the ardinalit y of F λ ∩ {⌊ S ⌋ λ , ⌈ S ⌉ λ } the new ativ e set A − λ has ℓ + 1 , ℓ , or ℓ − 1 elemen ts. More sp eially , if neither the left- nor the righ t-shifted soure no de w as a destination no de, then |A − λ | = ℓ + 1 . If either the left- or the righ t-shifted soure no de w as a destination no de, then |A − λ | = ℓ . If b oth the left- and righ t-shifted soure no de w ere destination no des, then |A − λ | = ℓ − 1 . In ea h of these ases A − λ is uniformly distributed among all subsets of M λ,η − 1 with ardinalit y A − λ that on tains no de 0 . Observ e that in all ases, the largest gap of A − λ is smaller or equal to the largest gap of A λ . W e also dene the follo wing transformations: MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 29 3 X =0 X X X 3 1 2 S reduce 1 X 2 X = =2 Λ Λ Figure A.2. Example of reduing for N = 16 , Λ = 4 . The sender is represen ted b y S and the no des of M 3 ha v e longer ti k marks. The no des of F 3 are irled. The no des ⌊ S ⌋ λ and ⌈ S ⌉ λ (as w ell as the 3 no des in b et w een) are merged in to the no de n um b ered 0 in the righ t illustration. Left (oun ter lo kwise) shifting: Sine S is uniformly distributed on { 1 , . . . , N } , the set (A.1) A ← λ := F λ ∪ {⌊ S ⌋ λ } is a random subset of M λ . W e an think of A ← λ as b eing hosen uniformly at random among all subsets of M λ ha ving ardinalit y |A ← λ | and sub jet to the same onditions as F λ . Notie that |A ← λ | = |F λ | if ⌊ S ⌋ λ ∈ F λ and |A ← λ | = |F λ | + 1 otherwise. 3 X X 1 S 4 =0 X X 2 3 X left shift 1 X 2 X = =2 =3 Λ Λ Λ Figure A.3. Example of left shifting for N = 16 , Λ = 4 . The destination no des are irled on the left, and the ativ e no des are irled on the righ t. The no des are ren um b ered after the shifting, starting with the former sender at 0. Also, the ativ e no des is ren um b ered, starting with X 1 > 0 , the rst ativ e no de after the former sender. The former sender is therefore the last ativ e no de, i.e., X 4 = 0 . Righ t (lo kwise) shifting: Analogously w e dene (A.2) A → λ := F λ ∪ {⌈ S ⌉ λ } . This is a random set hosen uniformly at random among all subsets of M λ ha ving ardinalit y |A → λ | and sub jet to the same onditions as F λ . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 30 3 X X 1 X 2 S 3 X =0 right shift 1 X 2 X 3 =2 = Λ Λ Λ Λ Figure A.4. Example of righ t shifting for N = 16 , Λ = 4 . After ren um b ering, the former sender is X 3 = 0 . Appendix B. Pr oof of Pr oposition 5.1 on Bounds f or Pr obability tha t CLG st ar ts a t Node 0 f or Hotspot Destina tion Traffi f or λ = Λ Pr o of. Conditioned on S ∈ M Λ , w e obtain (B.1) q ℓ β ( G Λ = 0 | S ∈ M Λ ) = 1 ℓ + 1 . Hene, w e only ha v e to onsider the ase S / ∈ M Λ . W e will not expliitly write do wn this ondition. Consider the righ t shifting and denote b y G → Λ the starting p oin t of the hosen largest gap of A → Λ . Sine N ≡ 0 is the only xed ativ e no de, the rst gap, i.e., { 0 , . . . , X Λ , 1 } , is the only one that nev er shrinks, while the last gap, i.e., { X Λ ,ℓ +1 , . . . , N } , is the only one that nev er gro ws. Therefore, q ℓ β ( G Λ = 0) ≤ q ℓ β ( G → Λ = 0) . (B.2) F or reasons of symmetry , w e ha v e q ℓ β ( G → Λ = 0 | ⌈ S ⌉ Λ / ∈ F Λ ) = q ℓ γ ( G Λ = 0) = 1 ℓ + 1 , (B.3) and q ℓ β ( G → Λ = 0 | ⌈ S ⌉ Λ ∈ F Λ ) = q ℓ − 1 γ ( G Λ = 0) = 1 ℓ . (B.4) The remaining probabilities an b e omputed as q ℓ β ( ⌈ S ⌉ Λ ∈ F Λ | S / ∈ M Λ ) = ℓ η , leading to the desired upp er b ound. Analogously , the left shifting yields a lo w er b ound, namely q ℓ β ( G Λ = 0 | ⌊ S ⌋ Λ 6 = 0) ≥ q ℓ β ( G ← Λ = 0 | ⌊ S ⌋ Λ 6 = 0) . (B.5) MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 31 Again for reasons of symmetry , w e obtain (B.6) q ℓ β ( G ← Λ = 0 | ⌊ S ⌋ Λ / ∈ F Λ ) = 1 ℓ + 1 and (B.7) q ℓ β ( G ← Λ = 0 | ⌊ S ⌋ Λ ∈ F Λ \ { 0 } ) = 1 ℓ . Finally , w e ha v e, of ourse, q ℓ β ( ⌊ S ⌋ Λ ∈ F Λ | S / ∈ M Λ ) = ℓ η and q ℓ β ( ⌊ S ⌋ Λ ∈ F Λ \ { 0 } | S / ∈ M Λ ) = ℓ − 1 . Appendix C. Pr oof of Theorem 6.1 on the Maximal Segment Utiliza tion Pr o of. Due to Equation (4.23), w e only ha v e to pro v e the ase of drop w a v elength Λ . Corollary 3.2 tell us that it sues to onsider the ritial segmen ts. Let n ≡ δ Λ with 1 ≤ δ < η b e a ritial segmen t for Λ . Analogously to the pro of of the domination priniple in [62℄, w e redue the domination priniple for hotsp ot destination tra to the statemen t (C.1) q ℓ β ( n ≥ G Λ > n − Λ) ≥ 1 η − δ q ℓ β ( G Λ > n − Λ) , and for hotsp ot soure tra to: (C.2) q ℓ γ ( G Λ = n ) ≥ 1 η − δ q ℓ γ ( G Λ ≥ n ) . 1 N Figure C.1. Illustration of statemen t (C.1): the mean slop e of a ertain p erio d is bigger or equal than the mean slop e o v er all later p erio ds In the γ (hotsp ot soure tra) setting, w e kno w that N is the sender, and th us A Λ ⊂ M Λ . Hene, w e do not need to onsider the no des on the other drop w a v elengths and the pro of is exatly the same as in the single w a v elength ase [ 62 ℄, see also gure C.2. W e will no w use the same strategy for the more ompliated pro of in the β (hotsp ot destination tra) setting. Let K n denote the n um b er of ativ e no des nding themselv es b et w een the no des N and n (lo kwise), i.e., (C.3) K n := |A Λ ∩ { 1 , . . . , n − Λ } | . MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 32 1 N Figure C.2. Gamma setting: the usage probabilit y sta ys onstan t on non ritial edges F or k ∈ { 0 , . . . , ( n − 1) ∧ ( ℓ − 1) } w e denote q ℓ,k γ for the probabilit y measure q ℓ γ onditioned on K n = k . W e denote again n ≡ δ Λ for δ ∈ { 1 , . . . , η − 1 } . W e will sho w that (C.4) q ℓ β ( n − Λ < G Λ ≤ n ) ≥ 1 η − δ q ℓ β ( G Λ > n − Λ) . 1 N Figure C.3. Beta setting: the usage probabilit y hanges along ea h segmen t In ase that S ∈ M Λ w e an again use the pro of of the one w a v elength senario. This is also true if S ∈ { 1 , . . . , n − Λ } , sine w e do not laim an ything ab out these no des. Hene, w e only ha v e to in v estigate the ase S ∈ { n − Λ + 1 , . . . , N } \ M Λ . F rom no w on w e assume this to b e the ase. W e deomp ose the left hand side in to t w o parts, (C.5) q ℓ β ( n − Λ < G Λ ≤ n ) = q ℓ β ( G Λ = n ) + q ℓ β ( G Λ = S, n − Λ < S < n ) . F or the rst summand of (C.5 ), w e pro eed similarly to the ase of a single w a v elength, namely q ℓ,k β ( G Λ = n ) = q ℓ,k β ( G Λ = n, G Λ ≥ n, ⌊ S ⌋ Λ 6 = n, n ∈ F Λ ) = q ℓ,k β G Λ = n G Λ ≥ n, ⌊ S ⌋ Λ 6 = n, n ∈ F Λ × × q ℓ,k β G Λ ≥ n ⌊ S ⌋ Λ 6 = n, n ∈ F Λ × q ℓ,k β ( ⌊ S ⌋ Λ 6 = n, n ∈ F Λ ) . (C.6) MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 33 W e obtain q ℓ,k β ( G Λ = n | G ≥ n, ⌊ S ⌋ Λ 6 = n, n ∈ F Λ ) = q ℓ − k β ,N − n +Λ ( G Λ = 1 | ⌊ S ⌋ Λ 6 = 1 , 1 ∈ F Λ ) ≥ q ℓ − k β ,N − n +Λ ( G ← Λ = 1 | ⌊ S ⌋ Λ 6 = 1 , 1 ∈ F Λ ) . (C.7) This probabilit y an b e omputed preisely q ℓ − k β ,N − n +Λ ( G ← Λ = 1 | ⌊ S ⌋ Λ 6 = 1 , 1 ∈ F Λ ) = q ℓ − k − 1 γ ,N − n ( G Λ = 0) q ℓ − k β ,N − n +Λ ( ⌊ S ⌋ Λ / ∈ F Λ | ⌊ S ⌋ Λ 6 = 1 , 1 ∈ F Λ ) + + q ℓ − k − 2 γ ,N − n ( G Λ = 0) q ℓ − k β ,N − n +Λ ( ⌊ S ⌋ Λ ∈ F Λ | ⌊ S ⌋ Λ 6 = 1 , 1 ∈ F Λ ) = 1 ℓ − k 1 − ℓ − k − 1 η − δ + 1 ℓ − k − 1 ℓ − k − 1 η − δ = 1 ℓ − k 1 + 1 η − δ . (C.8) W e no w use the fat that, onditionally on S ∈ { n − Λ + 1 , . . . , N } \ M Λ , (C.9) q ℓ,k β ( ⌊ S ⌋ Λ 6 = n ∈ F Λ ) = q ℓ,k β ( ⌊ S ⌋ Λ 6 = n ) q ℓ,k β ( n ∈ F Λ ) and (C.10) q ℓ,k β G Λ ≥ n ⌊ S ⌋ Λ 6 = n ∈ F Λ = q ℓ,k β G Λ ≥ n n ∈ F Λ q ℓ,k β ( ⌊ S ⌋ Λ 6 = n ) . Hene, w e obtain with q ℓ,k β ( n ∈ F Λ ) = ℓ − k − 1 η − δ that q ℓ,k β ( G Λ = n ) ≥ 1 η − δ q ℓ,k β G Λ ≥ n n ∈ F Λ × 1 − 1 ℓ − k 1 + 1 η − δ . (C.11) F or the seond part of (C.5 ), w e obtain q ℓ,k β ( G Λ ∈ I δ \ n ) = q ℓ,k β ( G Λ = S, G Λ ≥ S, ⌈ S ⌉ Λ = n, n / ∈ F Λ ) = q ℓ,k β G Λ = S G Λ ≥ S, ⌈ S ⌉ Λ = n, n / ∈ F Λ × × q ℓ,k β G Λ ≥ S ⌈ S ⌉ Λ = n / ∈ F Λ q ℓ,k β ( ⌈ S ⌉ Λ = n, n / ∈ F Λ ) . (C.12) W e ha v e q ℓ,k β ( G Λ = S | G Λ ≥ S, ⌈ S ⌉ Λ = n, n / ∈ F Λ ) = q ℓ − k β ,N − n +Λ ( G Λ = S | ⌈ S ⌉ Λ = 1 , 1 / ∈ F Λ ) . (C.13) MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 34 No w, w e use that |A → Λ | = |F Λ + 1 | for ⌈ S ⌉ Λ / ∈ F Λ . Hene, w e obtain q ℓ − k β ,N − n +Λ ( G Λ = S | ⌈ S ⌉ Λ = 1 , 1 / ∈ F Λ ) ≥ q ℓ − k β ,N − n +Λ ( G → Λ = 1 | ⌈ S ⌉ Λ = 1 , 1 / ∈ F Λ ) = q ℓ − k − 1 γ ,N − n ( G Λ = 0) = 1 ℓ − k . (C.14) Note that, onditioned on S ∈ { n − Λ + 1 , . . . , N } \ M Λ , w e ha v e (C.15) q ℓ,k β ( ⌈ S ⌉ Λ = n / ∈ F Λ ) = q ℓ,k β ( ⌈ S ⌉ Λ = n ) q ℓ,k β ( n / ∈ F Λ ) and (C.16) q ℓ,k β G Λ ≥ S ⌈ S ⌉ Λ = n / ∈ F Λ = q ℓ,k β G Λ ≥ S ⌈ S ⌉ Λ = n q ℓ,k β ( n / ∈ F Λ ) . Summarizing, w e obtain, using q ℓ,k β ( ⌈ S ⌉ Λ = n ) = 1 η − δ , that q ℓ,k β ( G Λ ∈ I δ \ n ) ≥ 1 η − δ 1 ℓ − k q ℓ,k β G Λ ≥ S ⌈ S ⌉ Λ = n . (C.17) It remains to sho w that q ℓ,k β ( G Λ > n − Λ) ≤ 1 − 1 ℓ − k 1 + 1 η − δ × × q ℓ,k β ( G Λ ≥ n | n ∈ F Λ ) + + 1 ℓ − k 1 − ℓ − k − 1 η − δ × × q ℓ,k β ( G Λ ≥ S | ⌈ S ⌉ Λ = n, n / ∈ F Λ ) . (C.18) This an b e sho wn b y q ℓ,k β ( G Λ > n − Λ) = η − ( ℓ − k ) X i = δ q ℓ,k β ( G Λ ≥ i Λ | X k +1 = i Λ) q ℓ,k β ( X k +1 = i Λ) + + Λ − 1 X λ =1 q ℓ,k β ( G Λ ≥ i Λ − λ | X k +1 = i Λ − λ ) q ℓ,k β ( X k +1 = i Λ − λ ) ≤ 1 − 1 ℓ − k q ℓ,k β ( G Λ ≥ n | X k +1 = n ) + + 1 ℓ − k q ℓ,k β ( G Λ ≥ S | ⌈ S ⌉ Λ = n ) . (C.19) F or the last inequalit y , w e used that for i ∈ { δ, . . . , η − 1 } and λ ∈ { 0 , . . . , Λ − 1 } q ℓ,k β ( G Λ ≥ i Λ − λ | X k +1 = i Λ − λ ) ≤ q ℓ,k β ( G Λ ≥ n − λ | X k +1 = n − λ ) (C.20) MUL TICAST CAP A CITY OF OPTICAL WDM P A CKET RING F OR HOTSPOT TRAFFIC 35 and, for reasons of symmetry , (C.21) q ℓ,k β ( X k +1 ∈ F Λ ) = 1 − 1 ℓ − k . The last step w e need is a omparison of ( C.18 ) and (C.19 ). The only dierene arises, when b oth of the ev en ts, { n ∈ F Λ } and {⌈ S ⌉ Λ = n } , tak e plae. Then, (C.22) q ℓ,k β ( G Λ ≥ S | ⌈ S ⌉ Λ = n ∈ F Λ ) = q ℓ,k β ( G Λ ≥ n | ⌈ S ⌉ Λ = n ∈ F Λ ) . This o urs with probabilit y q ℓ,k β ( n ∈ F Λ | ⌈ S ⌉ Λ = n ) = ℓ − k − 1 η − δ and explains the additional fator in the deomp osition (C.18 ). A kno wledgement W e are grateful to Martin Herzog, formerly of EMT, INRS, and Ra vi Sesha hala of Arizona State Univ ersit y for assistane with the n umerial and sim ulation ev aluations. Referenes [1℄ F. Da vik, M. Yilmaz, S. 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